Page 1
Extreme M-quantiles as risk measures: From L1 to Lp optimization
Supplementary Material
Abdelaati Daouiap1q, Stephane Girardp2q & Gilles Stupflerp3q
p1q Toulouse School of Economics, University of Toulouse Capitole, France
p2q Team Mistis, Inria Grenoble Rhone-Alpes & LJK, Inovallee, 655, av. de l’Europe,
Montbonnot, 38334 Saint-Ismier cedex, France
p3q School of Mathematical Sciences, University of Nottingham, University Park,
Nottingham NG7 2RD, United Kingdom
Supplement A contains the proofs of all theoretical results in the main paper. Supplement B provides additional
technical results and lemmas. Further simulation results are discussed in Supplement C. An application to medical
insurance data is given in Supplement D.
A Main results and proofs
Proof of Proposition 1. The starting point is that qτ ppq is a solution of the equation
p1´ τqEppq ´Xqp´11ItXăquq “ τEppX ´ qqp´11ItXąquq, (A.1)
which is equivalent to:
I1pq; pq “ p1´ τqI2pq; pq (A.2)
with I1pq; pq “ E
˜
„
X
q´ 1
p´1
1ItXąqu
¸
and I2pq; pq “ E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1¸
.
We now claim that τ ÞÑ qτ ppq is an increasing function on p0, 1q, tending to `8 as τ Ò 1. If indeed τ ÞÑ qτ ppq
were not an increasing function, one could find 0 ă τ1 ă τ2 ă 1 with qτ1ppq ě qτ2ppq. But then, since the maps
q ÞÑ Eppq ´ Xqp´11ItXăquq and q ÞÑ EppX ´ qqp´11ItXąquq are respectively nondecreasing and nonincreasing, one
would get thanks to (A.1) that:
p1´ τ2qEppqτ2ppq ´Xqp´11ItXăqτ2 ppquq ă p1´ τ1qEppqτ1ppq ´Xqp´11ItXăqτ1 ppquq
“ τ1EppX ´ qτ1ppqqp´11ItXąqτ1 ppquq ă τ2EppX ´ qτ2ppqqp´11ItXąqτ2 ppquq.
This is certainly a contradiction because of (A.1) again. Now, if qτ ppq did not tend to `8 as τ Ò 1, then it would
converge to some finite q˚ due to the function τ ÞÑ qτ ppq being increasing. The functions q ÞÑ Eppq´Xqp´11ItXăquq
and q ÞÑ EppX´ qqp´11ItXąquq being continuous on R by the dominated convergence theorem, this entails by letting
1
Page 2
τ Ò 1 in (A.1) with q ” qτ ppq that EppX ´ q˚qp´11ItXąq˚uq “ 0. Consequently X ď q˚ with probability 1, which is
a contradiction since X has a heavy right-tail and thus an infinite right endpoint.
The idea is then to compute asymptotic equivalents of both the expectations I1pq; pq and I2pq; pq as q Ñ `8 and
then solve equation (A.2) by replacing these terms by the aforementioned equivalents with qτ ppq substituted in
place of q.
We start by computing an asymptotic equivalent of I1pq; pq. Write
I1pq; pq “ Eˆ„
H
ˆ
X
q
˙
´Hp1q
1ItXąqu
˙
with Hpxq “ px´ 1qp´11Itxě1u, and apply Lemma 1(i) with b “ 1 to get
I1pq; pq “ F pqq
ż `8
1
pp´ 1qpx´ 1qp´2x´1{γdxp1` op1qq.
An integration by parts and the change of variables y “ 1{x entail
I1pq; pq “F pqq
γ
ż `8
1
px´ 1qp´1x´1{γ´1dxp1` op1qq
“F pqq
γ
ż 1
0
p1´ yqp´1y1{γ´pdyp1` op1qq
“F pqq
γBpp, γ´1 ´ p` 1qp1` op1qq (A.3)
as q Ñ `8.
We now examine I2pq; pq. Write
I2pq; pq “ I1pq; pq ` E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1ItXă´qu
¸
` E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1It|X|ďqu
¸
. (A.4)
By (A.3), the first term on the right-hand side above converges to 0 as q Ñ `8. The second one is controlled by
writing
E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1ItXă´qu
¸
ď
ˆ
2
q
˙p´1
E´
Xp´1´ 1ItXă´qu
¯
“ opq´pp´1qq “ op1q (A.5)
as q Ñ `8, while the asymptotic behavior of the third term is obtained by noting that the integrand converges
almost surely to 1 and is bounded by 2p´1 which, by the dominated convergence theorem, entails:
E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1It|X|ďqu
¸
Ñ 1 as q Ñ `8. (A.6)
Combining (A.3), (A.4), (A.5) and (A.6), we arrive at
I2pq; pq Ñ 1 as q Ñ `8. (A.7)
Using (A.3) and (A.7), equation (A.2) thus yields
1´ τ “ F pqτ ppqqBpp, γ´1 ´ p` 1q
γp1` op1qq
as τ Ò 1, which is the desired result.
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Proof of Proposition 2. As in the proof of Proposition 1, the starting point is the fact that qτ ppq is the unique
solution of equation (A.2). Let us provide an asymptotic expansion of both sides of this equation as τ Ò 1.
The left-hand side of (A.2) is the easiest part: we use Lemma 1(ii) with Hpxq “ px´ 1qp´11Itxě1u and b “ 1 to get,
as q Ñ `8,
I1pq; pq
F pqq´Bpp, γ´1
r ´ p` 1q
γr“ A
ˆ
1
F pqq
˙ż `8
1
pp´ 1qpx´ 1qp´2x´1{γrxρ{γr ´ 1
γrρdxp1` op1qq. (A.8)
When ρ ă 0, an integration by parts and the change of variables y “ 1{x entail
I1pq; pq
F pqq´Bpp, γ´1
r ´ p` 1q
γr
“ A
ˆ
1
F pqq
˙
1
γrρ
„
1´ ρ
γrBpp, p1´ ρqγ´1
r ´ p` 1q ´1
γrBpp, γ´1
r ´ p` 1q
p1` op1qq (A.9)
as q Ñ `8.
Let us now turn to the right-hand side of (A.2), which we break down as:
E
˜
ˇ
ˇ
ˇ
ˇ
X
q´ 1
ˇ
ˇ
ˇ
ˇ
p´1¸
“ I1pq; pq ` E
˜
„
1´X
q
p´1
1ItXďqu
¸
. (A.10)
An equivalent of I1pq; pq is already known by (A.3):
I1pq; pq “Bpp, γ´1
r ´ p` 1q
γrF pqqp1` op1qq
as q Ñ `8. The second term in (A.10) can be decomposed itself as follows:
E
˜
„
1´X
q
p´1
1ItXďqu
¸
“ E
˜#
„
1´X
q
p´1
´ 1
+
1ItXďqu
¸
` F pqq
“ J1pq; pq ` J2pq; pq ` 1´ F pqq (A.11)
with J1pq; pq “ E
˜#
„
1´X
q
p´1
´ 1
+
1It0ăXďqu
¸
and J2pq; pq “ E
˜#
„
1´X
q
p´1
´ 1
+
1ItXă0u
¸
.
We start by examining the asymptotic behavior of J1pq; pq. Let Hpxq “ ´pp ´ 1q´1p1 ´ xqp´11It0ďxď1u and apply
Lemma 1(iii), (iv) and (v) to obtain:
J1pq; pq “ ´pp´ 1qEˆ„
H
ˆ
X
q
˙
´Hp0q
1ItXą0u
˙
“ ´pp´ 1q
$
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
%
EpX1ItXą0uq
qp1` op1qq
if γr ă 1
or γr “ 1 and EpX`q ă 8,
EpX1It0ăXăquq
qp1` op1qq if γr “ 1 and EpX`q “ 8,
F pqqBpp´ 1, 1´ γ´1r qp1` op1qq if γr ą 1,
(A.12)
as q Ñ `8. To control J2pq; pq, notice first that
J2pq; pq “ E
˜#
„
1´X
q
p´1
´ 1
+
1ItXă0u
¸
“ E
˜#
„
1`X´q
p´1
´ 1
+
1ItX´ą0u
¸
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and apply Lemma 1(iii), (iv) and (v) with Hpxq “ pp´ 1q´1p1` xqp´1 to get
J2pq; pq “ pp´ 1qEˆ„
H
ˆ
X´q
˙
´Hp0q
1ItX´ą0u
˙
“ pp´ 1q
$
’
’
’
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
’
’
’
%
´EpX1ItXă0uq
qp1` op1qq
if γ` ă 1
or γ` “ 1 and EpX´q ă 8
or F´ is light-tailed,
´EpX1It´qăXă0uq
qp1` op1qq if γ` “ 1 and EpX´q “ 8,
F p´qqBpγ´1` ´ p` 1, 1´ γ´1
` qp1` op1qq if γ` ą 1.
(A.13)
This is obtained by noticing that, in the case γ` “ 1, we have EpX´1It0ăX´ăquq “ ´EpX1It´qăXă0uq, and in the
case γ` ą 1, the change of variables u “ x{p1` xq, or equivalently x “ u{p1´ uq, yields
ż `8
0
p1` xqp´2x´1{γ`dx “
ż 1
0
p1´ uq1{γ`´pu´1{γ`du “ Bpγ´1` ´ p` 1, 1´ γ´1
` q.
Finally, notice that the regular variation property of A (see Theorem 2.3.3 in de Haan and Ferreira, 2006) and
Proposition 1 entail
A
ˆ
1
F pqτ ppqq
˙
“
„
Bpp, γ´1r ´ p` 1q
γr
ρ
A
ˆ
1
1´ τ
˙
p1` op1qq. (A.14)
Combining (A.2), (A.9)–(A.14) and replacing q by qτ ppq shows that
F pqτ ppqq
ˆ
Bpp, γ´1r ´ p` 1q
γr`A
ˆ
1
1´ τ
˙
Kpp, γr, ρqp1` op1qq
˙
“ p1´ τq`
1´ F pqτ ppqq ´ pp´ 1qrRrpqτ ppq, p, γrq ´R`pqτ ppq, p, γ`qs˘
. (A.15)
Using Corollary 1 and the regular variation of the functions F and F´ (when it is heavy-tailed), we get
Rrpqτ ppq, p, γrq “
$
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
%
EpX1ItXą0uq
qτ ppqp1` op1qq
if γr ă 1
or γr “ 1 and EpX`q ă 8,
EpX1It0ăXăqτ ppquq
qτ ppqp1` op1qq if γr “ 1 and EpX`q “ 8,
F pqτ ppqqBpp´ 1, 1´ γ´1r qp1` op1qq if γr ą 1
“
$
’
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
’
%
„
γr
Bpp, γ´1r ´ p` 1q
γr EpX1ItXą0uq
qτ p1qp1` op1qq
if γr ă 1
or γr “ 1 and EpX`q ă 8,„
γr
Bpp, γ´1r ´ p` 1q
γr EpX1It0ăXăqτ p1quq
qτ p1qp1` op1qq if γr “ 1 and EpX`q “ 8,
γr
Bpp, γ´1r ´ p` 1q
F pqτ p1qqBpp´ 1, 1´ γ´1r qp1` op1qq if γr ą 1
„
„
γr
Bpp, γ´1r ´ p` 1q
minpγr,1q
Rrpqτ p1q, p, γrq
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Page 5
and
R`pqτ ppq, p, γ`q “
$
’
’
’
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
’
’
’
%
´EpX1ItXă0uq
qτ ppqp1` op1qq
if γ` ă 1
or γ` “ 1 and EpX´q ă 8
or F´ is light-tailed,
´EpX1It´qτ ppqăXă0uq
qτ ppqp1` op1qq if γ` “ 1 and EpX´q “ 8,
F p´qτ ppqqBpγ´1` ´ p` 1, 1´ γ´1
` qp1` op1qq if γ` ą 1
“
$
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
’
%
´
„
γr
Bpp, γ´1r ´ p` 1q
γr EpX1ItXă0uq
qτ p1qp1` op1qq
if γ` ă 1
or γ` “ 1 and EpX´q ă 8
or F´ is light-tailed,
´
„
γr
Bpp, γ´1r ´ p` 1q
γr EpX1It´qτ p1qăXă0uq
qτ p1qp1` op1qq if γ` “ 1 and EpX´q “ 8,
„
γr
Bpp, γ´1r ´ p` 1q
γr{γ`
ˆF p´qτ p1qqBpγ´1` ´ p` 1, 1´ γ´1
` qp1` op1qq if γ` ą 1
„
„
γr
Bpp, γ´1r ´ p` 1q
γr{maxpγ`,1q
R`pqτ p1q, p, γ`q.
Consequently, by Proposition 1,
F pqτ ppqq ` pp´ 1qrRrpqτ ppq, p, γrq ´R`pqτ ppq, p, γ`qs
“γr
Bpp, γ´1r ´ p` 1q
p1´ τqp1` op1qq
`pp´ 1q
˜
„
γr
Bpp, γ´1r ´ p` 1q
minpγr,1q
Rrpqτ p1q, p, γrq ´
„
γr
Bpp, γ´1r ´ p` 1q
γr{maxpγ`,1q
R`pqτ p1q, p, γ`q
¸
.
Rearranging equation (A.15) yields
F pqτ ppqq
1´ τ“
γr
Bpp, γ´1r ´ p` 1q
ˆ
1`A
ˆ
1
1´ τ
˙
γr
Bpp, γ´1r ´ p` 1q
Kpp, γr, ρqp1` op1qq
˙´1
ˆ
„
1´γr
Bpp, γ´1r ´ p` 1q
p1´ τqp1` op1qq
´pp´ 1q
˜
„
γr
Bpp, γ´1r ´ p` 1q
minpγr,1q
Rrpqτ p1q, p, γrq
´
„
γr
Bpp, γ´1r ´ p` 1q
γr{maxpγ`,1q
R`pqτ p1q, p, γ`q
¸ff
.
Using a straightforward Taylor expansion of the function x ÞÑ p1`xq´1 in a neighborhood of 0 completes the proof.
Proof of Proposition 3. By Proposition 2 and a Taylor expansion,
1´ τ
F pqτ ppqq“Bpp, γ´1
r ´ p` 1q
γrp1´Rpτ, pqp1` op1qqq.
Because Up1{p1´ τqq “ qτ p1q, the assertion is then a straightforward consequence of Lemma 2.
5
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Proof of Theorem 1. Notice that y ÞÑ ητ py; pq{p is continuously differentiable with derivative
ϕτ py; pq “ |τ ´ 1Ityď0u||y|p´1 signpyq.
Use Lemma 3 to write, for any u,
ψnpu; pq “ ´uT1,n ` T2,npuq ` T3,npuq (A.16)
with T1,n :“1
a
np1´ τnq
nÿ
i“1
1
rqτnppqsp´1
ϕτnpXi ´ qτnppq; pq,
T2,npuq :“ ´1
rqτnppqsp
nÿ
i“1
ż uqτn ppq{?np1´τnq
0
rEpϕτnpXi ´ qτnppq ´ t; pqq ´ EpϕτnpXi ´ qτnppq; pqqsdt
and T3,npuq :“ ´1
rqτnppqsp
nÿ
i“1
ż uqτn ppq{?np1´τnq
0
rSn,ipqτnppq ` tq ´ Sn,ipqτnppqqsdt
where Sn,ipvq :“ ϕτnpXi ´ v; pq ´ EpϕτnpX ´ v; pqq.
By Lemmas 8, 9 and 10, we get
ψnpu; pqdÝÑ ´uZ
a
V pγ; pqp1` σ2q `u2
2γas nÑ8
(with Z being standard Gaussian) in the sense of finite-dimensional convergence. As a function of u, this limit is
almost surely finite and defines a convex function which has a unique minimum at
u˚ “ γa
V pγ; pqp1` σ2qZd“ N
`
0, γ2V pγ; pqp1` σ2q˘
.
Applying the convexity lemma of Geyer (1996) completes the proof.
Proof of Theorem 2. Write
log
˜
pqWτ 1n ppq
qτ 1nppq
¸
“ ppγn ´ γq log
ˆ
1´ τn1´ τ 1n
˙
` log
ˆ
pqτnppq
qτnppq
˙
´ log
ˆ„
1´ τ 1n1´ τn
γ qτ 1nppq
qτnppq
˙
.
The convergence logrp1´ τnq{p1´ τ1nqs Ñ 8 yields
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ
pqτnppq
qτnppq
˙
“ OP`
1{ logrp1´ τnq{p1´ τ1nqs
˘
“ oPp1q, (A.17)
and
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ„
1´ τ 1n1´ τn
γ qτ 1nppq
qτnppq
˙
“
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqs
ˆ
log
ˆ
qτ 1nppq
qτ 1np1q
˙
´ log
ˆ
qτnppq
qτnp1q
˙
` log
ˆ„
1´ τ 1n1´ τn
γ qτ 1np1q
qτnp1q
˙˙
“ O
˜
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqsrRpτn, pq ` |App1´ τnq
´1q| `Rpτ 1n, pq ` |App1´ τ1nq´1q|s
¸
“ O
˜
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqsrRpτn, pq ` |App1´ τnq
´1q|s
¸
“ op1q. (A.18)
Convergence (A.17) is a consequence of our Theorem 1. Convergence (A.18) follows from a combination of
Proposition 3 and of Theorem 2.3.9 in de Haan and Ferreira (2006) and, in what concerns the relationship
Rpτ 1n, pq “ OpRpτn, pqq, from the regular variation of F , F´, s ÞÑ Upsq “ q1´s´1p1q and |A|. Combining these
elements and using the Delta-method leads to the desired conclusion.
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Page 7
Proof of Theorem 3. We start by writing
log
˜
rqWτ 1n ppq
qτ 1nppq
¸
“ log
˜
pqWτ 1n p1q
qτ 1np1q
¸
` log
ˆ
Cppγn; pq
Cpγr; pq
˙
´ log
ˆ
qτ 1nppq
Cpγr; pqqτ 1np1q
˙
. (A.19)
To work on the first term on the right-hand side, note that
log
˜
pqWτ 1n p1q
qτ 1np1q
¸
“ ppγn ´ γq log
ˆ
1´ τn1´ τ 1n
˙
` log
ˆ
pqτnp1q
qτnp1q
˙
´ log
ˆ„
1´ τ 1n1´ τn
γ qτ 1np1q
qτnp1q
˙
.
Since pqτnp1q “ Xn´tnp1´τnqu,n, the convergence logrp1´ τnq{p1´ τ1nqs Ñ 8 and a use of Theorem 2.3.9 of de Haan
and Ferreira (2006) yield:
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ
pqτnp1q
qτnp1q
˙
“ OP`
1{ logrp1´ τnq{p1´ τ1nqs
˘
“ oPp1q,
and
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ„
1´ τ 1n1´ τn
γ qτ 1np1q
qτnp1q
˙
“ O
˜
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqs|App1´ τnq
´1q|
¸
“ op1q.
As a consequence:a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
˜
pqWτ 1n p1q
qτ 1np1q
¸
dÝÑ ζ. (A.20)
To conclude the proof, it is then enough to examine the behavior of the second and third term on the right-hand
side of Equation (A.19). First,
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ
Cppγn; pq
Cpγr; pq
˙
“ OP`
1{ logrp1´ τnq{p1´ τ1nqs
˘
“ oPp1q, (A.21)
because of thea
np1´ τnq´convergence of pγn and of the differentiability of the mapping x ÞÑ logCpx; pq at γr.
Second,
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqslog
ˆ
qτ 1nppq
Cpγr; pqqτ 1np1q
˙
“ OP
˜
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqsrRpτ 1n, pq ` |App1´ τ
1nq´1q|s
¸
“ OP
˜
a
np1´ τnq
logrp1´ τnq{p1´ τ 1nqsrRpτn, pq ` |App1´ τnq
´1q|s
¸
“ oPp1q, (A.22)
which follows from a combination of Proposition 3 and of Theorem 2.3.9 in de Haan and Ferreira (2006) and, in
what concerns the relationship Rpτ 1n, pq “ OpRpτn, pqq, from the regular variation of F , F´, s ÞÑ Upsq “ q1´s´1p1q
and |A|. Combining these elements and using the Delta-method leads to the desired conclusion.
Proof of Theorem 4. We write
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q´ 1 “
γrpγnˆ
B
ˆ
p,1
pγn´ p` 1
˙
B
ˆ
p,1
γr´ p` 1
˙ ˆ
p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
1´ τ 1npp, αn; 1q´ 1. (A.23)
Nowa
np1´ τnq
ˆ
γrpγn´ 1
˙
“1
pγnˆa
np1´ τnqpγr ´ pγnqdÝÑ ´
ζ
γr(A.24)
7
Page 8
by Slutsky’s lemma. Moreover, using the relationship
BB
Bypx, yq “
B
By
ˆ
ΓpxqΓpyq
Γpx` yq
˙
“ Bpx, yq pΨpyq ´Ψpx` yqq
where Ψpxq “ Γ1pxq{Γpxq is the digamma function, we obtain
d
dx
„
B
ˆ
p,1
x´ p` 1
˙
“ ´1
x2B
ˆ
p,1
x´ p` 1
˙„
Ψ
ˆ
1
x´ p` 1
˙
´Ψ
ˆ
1
x` 1
˙
.
The delta-method then yields
a
np1´ τnq
¨
˚
˚
˝
B
ˆ
p,1
pγn´ p` 1
˙
B
ˆ
p,1
γr´ p` 1
˙ ´ 1
˛
‹
‹
‚
“1
B
ˆ
p,1
γr´ p` 1
˙ ˆa
np1´ τnq
„
B
ˆ
p,1
pγn´ p` 1
˙
´B
ˆ
p,1
γr´ p` 1
˙
dÝÑ ´
ζ
γ2r
„
Ψ
ˆ
1
γr´ p` 1
˙
´Ψ
ˆ
1
γr` 1
˙
. (A.25)
To complete the proof, we note that
p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
1´ τ 1npp, αn; 1q“
p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
E
«
ˇ
ˇ
ˇ
ˇ
X
qαnp1q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1ItXąqαn p1qu
ffE
«
ˇ
ˇ
ˇ
ˇ
X
qαnp1q´ 1
ˇ
ˇ
ˇ
ˇ
p´1ff
.
Recall now (A.8) in the proof of Proposition 2 which here translates into
p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
E
«
ˇ
ˇ
ˇ
ˇ
X
qαnp1q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1ItXąqαn p1qu
ff ´ 1 “
F pqαnp1qq1
γrB
ˆ
p,1
γr´ p` 1
˙
E
«
ˇ
ˇ
ˇ
ˇ
X
qαnp1q´ 1
ˇ
ˇ
ˇ
ˇ
p´1
1ItXąqαn p1qu
ff ´ 1
“ OrAp1{F pqαnp1qqqs
“ OrApp1´ αnq´1qs.
Similarly, by (A.10)–(A.13) in the proof of Proposition 2, we get
E
«
ˇ
ˇ
ˇ
ˇ
X
qαnp1q´ 1
ˇ
ˇ
ˇ
ˇ
p´1ff
´ 1 “ OrmaxtF pqαnp1qq, Rrpqαnp1q, p, γrq, R`pqαnp1q, p, γ`qus
“ Ormaxt1´ αn, Rrpqαnp1q, p, γrq, R`pqαnp1q, p, γ`qus.
Combine these two asymptotic bounds to obtain
p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
1´ τ 1npp, αn; 1q´ 1 “ Ormaxt1´ αn, App1´ αnq
´1q, Rrpqαnp1q, p, γrq, R`pqαnp1q, p, γ`qus. (A.26)
Combining (A.23), (A.24), (A.25) and (A.26) leads to
a
np1´ τnq
ˆ
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q´ 1
˙
“ ´
"
1`1
γr
„
Ψ
ˆ
1
γr´ p` 1
˙
´Ψ
ˆ
1
γr` 1
˙*
ζ
γr`Op1q “ OPp1q
8
Page 9
proving the first statement. In the case when
a
np1´ τnqmaxt1´ αn, App1´ αnq´1q, Rrpqαnp1q, p, γrq, R`pqαnp1q, p, γ`qu Ñ 0
the above equality becomes
a
np1´ τnq
ˆ
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q´ 1
˙
“ ´
"
1`1
γr
„
Ψ
ˆ
1
γr´ p` 1
˙
´Ψ
ˆ
1
γr` 1
˙*
ζ
γr` op1q
which implies the second statement and concludes the proof.
Proof of Theorem 5. The key point is to write
pqWpτ 1npp,αn;1q
ppq “
ˆ
1´ pτ 1npp, αn; 1q
1´ τn
˙´pγn
pqτnppq “
ˆ
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q
˙´pγn
ˆ
#
ˆ
1´ τ 1npp, αn; 1q
1´ τn
˙´pγn
pqτnppq
+
. (A.27)
Now, by Theorem 4,
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q“ 1`OP
˜
1a
np1´ τnq
¸
and thereforeˆ
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q
˙´pγn
“ exp
ˆ
´pγn log
„
1´ pτ 1npp, αn; 1q
1´ τ 1npp, αn; 1q
˙
“ exp
˜
´
«
γ `OP
˜
1a
np1´ τnq
¸ff
ˆOP
˜
1a
np1´ τnq
¸¸
“ 1`OP
˜
1a
np1´ τnq
¸
(A.28)
by a Taylor expansion. Furthermore, we haveˆ
1´ τ 1npp, αn; 1q
1´ τn
˙´pγn
pqτnppq “ pqWτ 1npp,αn;1qppq
by definition of the extrapolated class of estimators pqW ppq. Using the asymptotic equivalent
1´ τ 1npp, αn; 1q „ p1´ αnq1
γrB
ˆ
p,1
γr´ p` 1
˙
(A.29)
we conclude that the conditions of Theorem 2 are satisfied if the parameter τ 1n there is set equal to τ 1npp, αn; 1q. By
Theorem 2:a
np1´ τnq
logrp1´ τnq{p1´ τ 1npp, αn; 1qqs
˜
pqWτ 1npp,αn;1qppq
qτ 1npp,αn;1qppq´ 1
¸
dÝÑ ζ.
Now
log
„
1´ τn1´ τ 1npp, αn; 1q
“ log
„
1´ τn1´ αn
` log
„
1´ αn1´ τ 1npp, αn; 1q
and in the right-hand side of this identity, the first term tends to infinity, while the second term converges to a
finite constant in view of (A.29). As a conclusion
log
„
1´ τn1´ τ 1npp, αn; 1q
„ log
„
1´ τn1´ αn
.
Together with the equality qτ 1npp,αn;1qppq “ qαnp1q which is true by definition of τ 1npp, αn; 1q, this entailsa
np1´ τnq
logrp1´ τnq{p1´ αnqs
˜
pqWτ 1npp,αn;1qppq
qαnp1q´ 1
¸
dÝÑ ζ. (A.30)
Combining (A.27), (A.28) and (A.30) completes the proof of the first convergence.
9
Page 10
Proof of Theorem 6. The proof of this result is similar to that of Theorem 5: just apply Theorem 3 instead of
Theorem 2 in order to prove the required analogue of (A.30).
Proof of Theorem 7. The proof of this result is the same as that of Theorem 3, with pqWτ 1n p1q being replaced by
pqWτ 1n ppq [thus applying Theorem 2 to obtain an analogue of (A.20)] and the mapping x ÞÑ logCpx; pq being replaced
by x ÞÑ logrCpx; 2qC´1px; pqs. The details of the proof are therefore omitted.
Proof of Theorem 8. The proof of this result is entirely similar to that of Theorem 5 and is therefore omitted.
Proof of Theorem 9. The proof of this result is entirely similar to that of Theorem 6 and is therefore omitted.
B Auxiliary results and proofs
Lemma 1. Let X be a random variable whose survival function F satisfies condition C1pγq, and let H be an
absolutely continuous function whose derivative h is nonnegative and is such that
Da ě 0, Dδ ą 0, @b ą a,
ż `8
b
hpxqx´1{γ`δdx ă 8.
(i) For any b ą a, we have, as q Ñ `8:
Eˆ„
H
ˆ
X
q
˙
´Hpbq
1ltXąbqu
˙
“ F pqq
ż `8
b
hpxqx´1{γdxp1` op1qq.
(ii) If moreover F satisfies condition C2pγ, ρ,Aq, then for any b ą a, we have, as q Ñ `8:
Eˆ„
H
ˆ
X
q
˙
´Hpbq
1ltXąbqu
˙
“ F pqq
ˆż `8
b
hpxqx´1{γdx`A
ˆ
1
F pqq
˙ż `8
b
hpxqx´1{γ xρ{γ ´ 1
γρdxp1` op1qq
˙
.
Assume further that a “ 0 and that h is right-continuous at 0 with hp0q “ 1. Let X` “ maxpX, 0q denote the
positive part of X.
(iii) If γ ă 1, or γ “ 1 and EpX`q ă 8, then, as q Ñ `8:
Eˆ„
H
ˆ
X
q
˙
´Hp0q
1ltXą0u
˙
“EpX`qq
p1` op1qq.
This result also holds true if the function F is actually light-tailed, provided there is c ą 0 such that the
function x ÞÑ x´chpxq is integrable in a neighborhood of infinity.
(iv) If γ “ 1 and EpX`q “ 8, then the function q ÞÑ EpX1lt0ăXăquq is slowly varying and, as q Ñ `8:
Eˆ„
H
ˆ
X
q
˙
´Hp0q
1ltXą0u
˙
“EpX1lt0ăXăquq
qp1` op1qq.
(v) If γ ą 1, then, as q Ñ `8:
Eˆ„
H
ˆ
X
q
˙
´Hp0q
1ltXą0u
˙
“ F pqq
ż `8
0
hpxqx´1{γdxp1` op1qq.
10
Page 11
Proof of Lemma 1. The basic idea of the proof is to note that an integration by parts entails, for b ą a:
Ipb; qq :“ Eˆ„
H
ˆ
X
q
˙
´Hpbq
1ltXąbqu
˙
“
ż `8
b
hpxqF pqxqdx.
To show (i), write
Ipb; qq “ F pqq
ˆż `8
b
hpxqx´1{γdx`
ż `8
b
hpxq
„
F pqxq
F pqq´ x´1{γ
dx
˙
(B.1)
and use a uniform bound such as Theorem B.2.18 in de Haan and Ferreira (2006) to get
Ipb; qq “ F pqq
ż `8
b
hpxqx´1{γdxp1` op1qq
as q Ñ `8, which is (i).
Assertion (ii) is obtained in a similar way by using (B.1), the second-order condition C2pγ, ρ,Aq and a uniform
inequality such as Theorem B.3.10 in de Haan and Ferreira (2006) applied to the function F .
The first step in order to show (iii), (iv) and (v) is to split Ip0; qq as
Ip0; qq “
ż ε
0
hpxqF pqxqdx`
ż `8
ε
hpxqF pqxqdx
“1
q
ż qε
0
h
ˆ
x
q
˙
F pxqdx`
ż `8
ε
hpxqF pqxqdx (B.2)
where ε is an arbitrary positive real number. To prove (iii), note that if X ď 0 almost surely there is nothing to
prove; otherwise, because
EpX1lt0ăXăqεuq “
ż qε
0
F pxqdx´ qεF pqεq,
we obtain:
Ip0; qq ´EpX1lt0ăXăqεuq
q“
1
q
ż qε
0
„
h
ˆ
x
q
˙
´ 1
F pxqdx`
ż `8
ε
hpxqF pqxqdx`1
qtqεF pqεqu.
Since EpX`q ă 8 the function F is nonincreasing and integrable in a neighborhood of infinity. This entails
xF pxq ď 2
ż x
x{2
F ptqdtÑ 0 as xÑ `8
and therefore that F pqq “ op1{qq as q Ñ `8; this is of course also true if F is light-tailed. We thus obtain, by
representation (B.1) when F is regularly varying:
@ε ą 0, Ip0; qq ´EpX1lt0ăXăqεuq
q“
1
q
ż qε
0
„
h
ˆ
x
q
˙
´ 1
F pxqdx` o
ˆ
1
q
˙
.
By the dominated convergence theorem, EpX1ltXěqεuq Ó 0 as q Ñ `8 and then:
@ε ą 0, Ip0; qq ´EpX`qq
“1
q
ż qε
0
„
h
ˆ
x
q
˙
´ 1
F pxqdx` o
ˆ
1
q
˙
.
For a given α ą 0, choose now ε such that |hpxq ´ 1| ď α{p1` EpX`qq for all x P r0, εs; this yields
ˇ
ˇ
ˇ
ˇ
Ip0; qq ´EpX`qq
ˇ
ˇ
ˇ
ˇ
ďα
1` EpX`q
"EpX1lt0ăXăqεuq
q
*
`α
1` EpX`q1
qďα
q
for q large enough. Because α is arbitrary, this completes the proof of (iii).
11
Page 12
To show (iv), use (B.2) to get for any ε ą 0:
Ip0; qq “1
q
ż 1
0
h
ˆ
x
q
˙
F pxqdx`
ż ε
1{q
hpxqF pqxqdx`
ż `8
ε
hpxqF pqxqdx
for q large enough. By the right-continuity of h at 0 and part (i) of the present Lemma, we get
Ip0; qq “
ż ε
1{q
hpxqF pqxqdx`O
ˆ
max
„
1
q, F pqq
˙
.
For an arbitrary α P p0, 1q, choose now ε so small that hpxq P r1´ α{4, 1` α{4s when x P r0, εs. We get
´
1´α
4
¯
ż ε
1{q
F pqxqdx ď
ż ε
1{q
hpxqF pqxqdx ď´
1`α
4
¯
ż ε
1{q
F pqxqdx
ô
´
1´α
4
¯ 1
q
ż qε
1
F pxqdx ď
ż ε
1{q
hpxqF pqxqdx ď´
1`α
4
¯ 1
q
ż qε
1
F pxqdx.
By Proposition 1.5.9a in Bingham et al. (1987), the function z ÞÑşz
1F pxqdx “
şz
1txF pxqudx{x is slowly varying in
a neighborhood of `8 (i.e. regularly varying with index 0) so that for q large enough,
´
1´α
2
¯ 1
q
ż q
1
F pxqdx ď
ż ε
1{q
hpxqF pqxqdx ď´
1`α
2
¯ 1
q
ż q
1
F pxqdx.
Finally, we haveşq
1F pxqdx Ò EppX ´ 1q1ltXą1uq “ `8 as q Ñ `8 and, by Proposition 1.5.9a in Bingham et al.
(1987):1
F pqq
"
1
q
ż q
1
F pxqdx
*
“1
qF pqq
ż q
1
txF pxqudx
xÑ `8 (B.3)
as q Ñ `8. In other words, for q large enough,
p1´ αq1
q
ż q
1
F pxqdx ď Ip0; qq ď p1` αq1
q
ż q
1
F pxqdx.
Since α is arbitrary, this entails
Ip0; qq “1
q
ż q
1
F pxqdxp1` op1qq “1
q
ż q
0
F pxqdxp1` op1qq
as q Ñ `8. Finally, by (B.3),
EpX1lt0ăXăquq “
ż q
0
F pxqdx´ qF pqq “
ż q
0
F pxqdx
˜
1´qF pqq
şq
0F pxqdx
¸
“
ż q
0
F pxqdxp1` op1qq
so that
Ip0; qq “EpX1lt0ăXăquq
qp1` op1qq
as q Ñ `8; the proof of (iv) is then complete.
To show (v), let β P p0, 1q be such that 1{γ ă 1´ β and use once again (B.2) to get:
Ip0; qq “1
q
ż qβ
0
h
ˆ
x
q
˙
F pxqdx`
ż `8
q´p1´βqhpxqF pqxqdx.
By the right-continuity of h at 0 and the asymptotic relationship q´p1´βq “ opF pqqq as q Ñ `8,
Ip0; qq “1
q
ż qβ
0
F pxqdxp1` op1qq `
ż `8
q´p1´βqhpxqF pqxqdx “
ż `8
q´p1´βqhpxqF pqxqdx` opF pqqq.
12
Page 13
In the spirit of the proof of (i), write now
Ip0; qq “ F pqq
ˆż `8
q´p1´βqhpxqx´1{γdx`
ż `8
q´p1´βqhpxq
„
F pqxq
F pqq´ x´1{γ
dx
˙
` opF pqqq.
Since ´1{γ ą ´1, the function x ÞÑ x´1{γ is integrable in a neighborhood of 0, and thus
Ip0; qq “ F pqq
ˆż `8
0
hpxqx´1{γdx`
ż `8
q´p1´βqhpxq
„
F pqxq
F pqq´ x´1{γ
dx
˙
` opF pqqq.
Finally, since in the second integral we have qx ě qβ Ñ `8, we may use again a uniform bound such as Theorem
B.2.18 in de Haan and Ferreira (2006) to get
Ip0; qq “ F pqq
ż `8
0
hpxqx´1{γdxp1` op1qq
as q Ñ `8, which completes the proof of (v).
Lemma 2. Assume that v, V are such that vpτq Ò 8 and V pτq Ó 0, as τ Ò 1, and there exists B ą 0 such that
V pτq
F pvpτqq“ Bp1` epτqq
where epτq Ñ 0 as τ Ò 1. If condition C2pγ, ρ,Aq holds, with γ ą 0 and F strictly increasing, then
vpτq
Up1{V pτqq“ Bγ
ˆ
1` γepτqp1` op1qq `Ap1{V pτqq
„
Bρ ´ 1
ρ` op1q
˙
as τ Ò 1.
Proof. Apply the function U to get
vpτq
Up1{V pτqq´Bγ “
UpBr1` epτqs{V pτqq
Up1{V pτqq´Bγ .
By Theorem 2.3.9 in de Haan and Ferreira (2006), we may find a function A0, equivalent to A at infinity, such that
for any ε ą 0, there is t0pεq ą 1 such that for t, tx ě t0pεq,
ˇ
ˇ
ˇ
ˇ
1
A0ptq
ˆ
Uptxq
Uptq´ xγ
˙
´ xγxρ ´ 1
ρ
ˇ
ˇ
ˇ
ˇ
ďε
rp2Bqγ`ρ ` pB{2qγ`ρsrp2Bqε ` pB{2q´εsxγ`ρ maxpxε, x´εq.
Thus, for τ sufficiently close to 1, using this inequality with t “ 1{V pτq and x “ Br1` epτqs gives that
ˇ
ˇ
ˇ
ˇ
1
A0p1{V pτqq
ˆ
UpBr1` epτqs{V pτqq
Up1{V pτqq´Bγp1` epτqqγ
˙
´Bγp1` epτqqγBρp1` epτqqρ ´ 1
ρ
ˇ
ˇ
ˇ
ˇ
ď ε
and therefore1
A0p1{V pτqq
ˆ
UpBr1` epτqs{V pτqq
Up1{V pτqq´Bγp1` epτqqγ
˙
Ñ BγBρ ´ 1
ρas τ Ò 1.
The desired result follows by a simple first-order Taylor expansion.
In the next result we use the fact that y ÞÑ ητ py; pq{p is continuously differentiable with derivative
ϕτ py; pq “ |τ ´ 1ltyď0u||y|p´1 signpyq.
Lemma 3. For all x, y P R and τ P p0, 1q,
1
ppητ px´ y; pq ´ ητ px; pqq “ ´yϕτ px; pq ´
ż y
0
pϕτ px´ t; pq ´ ϕτ px; pqqdt.
13
Page 14
Proof of Lemma 3. The result follows from the identity
1
ppητ px´ y; pq ´ ητ px; pqq “
ż x´y
x
ϕτ ps; pqds “ ´
ż y
0
ϕτ px´ t; pqdt
obtained by the change of variables s “ x´ t.
The next lemma gives asymptotic equivalents for a number of moments that will be used in our examination of the
convergence of the direct empirical estimator.
Lemma 4. Assume that the survival function F satisfies condition C1pγq. Pick a ě 1 and assume that γ ă
1{rapp´ 1qs and EpXapp´1q´ q ă 8. Then:
(i) We have
Ep|ϕτ pX´qτ ppq; pq|a1ltXąqτ ppquq “ app´1qrqτ ppqsapp´1qp1´τq
γBpapp´ 1q, γ´1 ´ app´ 1qq
Bpp, γ´1 ´ p` 1qp1`op1qq as τ Ò 1.
(ii) We have
Ep|ϕτ pX ´ qτ ppq; pq|a1ltXďqτ ppquq “ p1´ τqarqτ ppqs
app´1qp1` op1qq as τ Ò 1.
(iii) When a ą 1, we have
Ep|ϕτ pX ´ qτ ppq; pq|aq “ app´ 1qrqτ ppqsapp´1qp1´ τq
γBpapp´ 1q, γ´1 ´ app´ 1qq
Bpp, γ´1 ´ p` 1qp1` op1qq as τ Ò 1.
Proof of Lemma 4. Define θ “ app´ 1q. To show (i), note that
Ep|ϕτ pX ´ qτ ppq; pq|a1ltXąqτ ppquq “ τaEprX ´ qτ ppqsθ1ltXąqτ ppquq
and apply Lemma 1(i) with Hpxq “ px´ 1qθ1ltxě1u and b “ 1 to get
EprX ´ qτ ppqsθ1ltXąqτ ppquq “ θrqτ ppqsθF pqτ ppqq
ż 8
1
pv ´ 1qθ´1v´1{γdvp1` op1qq as τ Ò 1.
Combining this equality with Proposition 1 and the change of variables u “ 1´ v´1, we obtain
EprX ´ qτ ppqsθ1ltXąqτ ppquq “ θrqτ ppqsθp1´ τq
γBpθ, γ´1 ´ θq
Bpp, γ´1 ´ p` 1qp1` op1qq as τ Ò 1
which is (i). To show (ii), write
Ep|ϕτ pX ´ qτ ppq; pq|a1ltXďqτ ppquq “ p1´ τqarqτ ppqs
θE
˜
„
1´X
qτ ppq
θ
1ltXďqτ ppqu
¸
.
The conditions γ ă θ´1 and EpXθ´q ă 8 ensure that E|X|θ ă 8. Recall that qτ ppq Ò `8 as τ Ò 1 and use the
dominated convergence theorem to get
Ep|ϕτ pX ´ qτ ppq; pq|a1ltXďqτ ppquq “ p1´ τqarqτ ppqs
θp1` op1qq
as required. Finally, combining (i) and (ii) gives (iii) and concludes the proof.
14
Page 15
Lemma 5. Let pxnq be a positive sequence tending to infinity and pht,nq, t P Tn be a class of functions such that
suptPTn
supxěxn
|ht,npxq| Ñ 0 as nÑ8.
(i) Assume that the survival function F satisfies condition H1pγq. Then:
suptPTn
supxěxn
|ht,npxq|´1
ˇ
ˇ
ˇ
ˇ
F pxp1` ht,npxqqq
F pxq´
ˆ
1´ht,npxq
γ
˙ˇ
ˇ
ˇ
ˇ
Ñ 0 as nÑ8.
(ii) Assume that the survival function F satisfies condition C2pγ, ρ,Aq. Then:
suptPTn
supxěxn
“
maxp|ht,npxq|, |Ap1{F pxqq|q‰´1
ˇ
ˇ
ˇ
ˇ
F pxp1` ht,npxqqq
F pxq´
ˆ
1´ht,npxq
γ
˙ˇ
ˇ
ˇ
ˇ
Ñ 0 as nÑ8.
Proof of Lemma 5. We first prove (i). Write for any ph, xq:
F pxp1` hqq
F pxq“ p1` hq´1{γ cpxp1` hqq
cpxqexp
˜
ż xp1`hq
x
∆puq
udu
¸
. (B.4)
By the mean value theorem, we have for n large enough
|cpxp1` ht,npxqqq ´ cpxq| ď |xht,npxq| maxyPrx,xp1`ht,npxqqs
|c1pyq| ď 2|ht,npxq| maxyPrxn{2,8q
|yc1pyq|
for all t P Tn and x ě xn, which entails
suptPTn
supxěxn
1
|ht,npxq||cpxp1` ht,npxqqq ´ cpxq| Ñ 0 as nÑ8. (B.5)
Furthermore1
|ht,npxq|
ˇ
ˇ
ˇ
ˇ
ˇ
ż xp1`ht,npxqq
x
∆puq
udu
ˇ
ˇ
ˇ
ˇ
ˇ
ď
ˇ
ˇ
ˇ
ˇ
logp1` ht,npxqq
ht,npxq
ˇ
ˇ
ˇ
ˇ
maxyPrx,xp1`ht,npxqqs
|∆pyq| Ñ 0
as nÑ8 for all t P Tn and x ě xn, so that the inequality |ez ´ 1| ď |z|e|z| yields
suptPTn
supxěxn
1
|ht,npxq|
ˇ
ˇ
ˇ
ˇ
ˇ
exp
˜
ż xp1`ht,npxqq
x
∆puq
udu
¸
´ 1
ˇ
ˇ
ˇ
ˇ
ˇ
Ñ 0 as nÑ8. (B.6)
Combine (B.4), (B.5) and (B.6) with the Taylor expansion p1` hq´1{γ “ 1´ h{γ ` ophq as hÑ 0 to complete the
proof of (i).
We now turn to the proof of (ii). Using a uniform inequality such as Theorem B.3.10 in de Haan and Ferreira (2006)
applied to the function F , we get that for any ε ą 0 small enough there is x0 ą 1 such that for all x ě 2x0 and
s P r1{2, 2s:ˇ
ˇ
ˇ
ˇ
1
Ap1{F pxqq
„
F psxq
F pxq´ s´1{γ
ˇ
ˇ
ˇ
ˇ
ď ε.
Applying this to s “ 1` ht,npxq, x ě xn and letting εÑ 0 we obtain:
suptPTn
supxěxn
ˇ
ˇ
ˇ
ˇ
1
Ap1{F pxqq
„
F pxp1` ht,npxqqq
F pxq´ p1` ht,npxqq
´1{γ
ˇ
ˇ
ˇ
ˇ
“ op1q.
Using again the Taylor expansion p1` hq´1{γ “ 1´ h{γ ` ophq as hÑ 0 completes the proof.
The next result gives a Lipschitz property for the derivative ϕτ .
15
Page 16
Lemma 6. For all x, h P R and τ P p0, 1q, we have
ϕτ px´ h; pq ´ ϕτ px; pq “ |τ ´ 1ltxď0u|`
|x´ h|p´1 signpx´ hq ´ |x|p´1 signpxq˘
` p1´ 2τqp1ltxďhu ´ 1ltxď0uq|x´ h|p´1 signpx´ hq.
Especially,
|ϕτ px´ h; pq ´ ϕτ px; pq| ď |h|p´11lt|x|ď|h|u ` p1´ τ ` 1ltxą0uq
$
’
’
’
&
’
’
’
%
2|h|p´1 if 1 ă p ă 2
pp´ 1qp2p´2 ` 1qp|h|p´1 ` |x|p´2|h|q if p ě 2.
Proof of Lemma 6. The equality result is a straightforward consequence of the fact that
|τ ´ 1ltxďhu| ´ |τ ´ 1ltxď0u| “ p1´ τqp1ltxďhu ´ 1ltxď0uq ` τp1ltxąhu ´ 1ltxą0uq “ p1´ 2τqp1ltxďhu ´ 1ltxď0uq.
To show the bound on the oscillation of ϕτ , note first that
1ltxďhu ´ 1ltxď0u “
$
’
&
’
%
1lt0ăxďhu if h ą 0
´1lthăxď0u if h ă 0
and consequently
|p1´ 2τqp1ltxďhu ´ 1ltxď0uq|x´ h|p´1 signpx´ hq| ď
$
’
&
’
%
|x´ h|p´11lt0ăxďhu if h ą 0
|x´ h|p´11lthăxď0u if h ă 0
ď |h|p´11lt|x|ď|h|u. (B.7)
Next, when 1 ă p ă 2, because v ÞÑ vp´2 is decreasing on p0,8q it is clear that
||x´ h|p´1 signpx´ hq ´ |x|p´1 signpxq| “
ˇ
ˇ
ˇ
ˇ
ˇ
ż x´h
x
pp´ 1q|v|p´2dv
ˇ
ˇ
ˇ
ˇ
ˇ
ď pp´ 1q
ż |h|
´|h|
|v|p´2dv
“ 2|h|p´1. (B.8)
When p ě 2, write
||x´ h|p´1 signpx´ hq ´ |x|p´1 signpxq| “
ˇ
ˇ
ˇ
ˇ
ˇ
ż x´h
x
pp´ 1q|v|p´2dv
ˇ
ˇ
ˇ
ˇ
ˇ
ď pp´ 1q|h| maxvPrx,x´hs
|v|p´2
ď pp´ 1q|h|r|x´ h|p´2 ` |x|p´2s
by the monotonicity of v ÞÑ vp´2 on r0,8q, and therefore
||x´ h|p´1 signpx´ hq ´ |x|p´1 signpxq| ď pp´ 1q|h|rp|x| ` |h|qp´2 ` |x|p´2s
ď pp´ 1qp2p´2 ` 1q|h|rmaxp|x|, |h|qsp´2
ď pp´ 1qp2p´2 ` 1qp|h|p´1 ` |x|p´2|h|q. (B.9)
Combining (B.7), (B.8) and (B.9) completes the proof.
16
Page 17
The lemma below is a useful convergence result for the variance of row-wise partial sums of a triangular array of
strictly stationary, dependent and square-integrable random variables.
Lemma 7. Let pVi,jq be a triangular array of square-integrable random variables such that:
• for any positive integer n and any k ď n, the random variable Vn,k is σpXkq´measurable;
• for any positive integer n, the random variables Vn,k, 1 ď k ď n are identically distributed.
Then, if the sequence pXnq is ρ´mixing withř8
n“1 ρpnq ă 8, we have
limnÑ8
1
nVarpVn,1qVar
˜
nÿ
k“1
Vn,k
¸
exists and is finite.
Proof of Lemma 7. Use the strict stationarity of the sequence to obtain
Var
˜
nÿ
k“1
Vn,k
¸
“ nVarpVn,1q ` 2nÿ
k“2
pn´ k ` 1qCovpVn,1, Vn,kq
“ nVarpVn,1q
˜
1` 2nÿ
k“2
n´ k ` 1
ncorrpVn,1, Vn,kq
¸
.
It is therefore enough to show that the sequence psnq defined by
sn :“nÿ
k“2
n´ k ` 1
ncorrpVn,1, Vn,kq
converges, or equivalently, that it is a Cauchy sequence. For this, we use the definition of the mixing coefficients
ρpnq to obtain, for any positive integers p and q,
|sp ´ sp`q| ď
pÿ
k“2
ˇ
ˇ
ˇ
ˇ
p´ k ` 1
p´p` q ´ k ` 1
p` q
ˇ
ˇ
ˇ
ˇ
ρpk ´ 1q `p`qÿ
k“p`1
p` q ´ k ` 1
p` qρpk ´ 1q
“q
p` q
pÿ
k“2
k ´ 1
pρpk ´ 1q `
p`qÿ
k“p`1
p` q ´ k ` 1
p` qρpk ´ 1q
ď1
p
p´1ÿ
k“1
kρpkq `p`q´1ÿ
k“p
ρpkq.
Kronecker’s lemma gives that the first sum above is arbitrarily small for p large enough due to the convergence of
the seriesř8
n“1 ρpnq; besides, the second term is less than a remainder of this convergent series starting at the pth
term, and is therefore arbitrarily small as well for p large enough. Consequently |sp ´ sp`q| is arbitrarily small if p
is chosen large enough, which entails the convergence of psnq and concludes the proof.
The last three results are the essential steps to the proof of Theorem 1.
Lemma 8. Work under the conditions of Theorem 1. Let
T1,n “1
a
np1´ τnq
nÿ
i“1
1
rqτnppqsp´1
ϕτnpXi ´ qτnppq; pq.
Then there is σ2 P r0,8q such that
T1,ndÝÑ N
`
0, V pγ; pqp1` σ2q˘
as nÑ8.
If moreover pXnq is an independent sequence, then σ2 “ 0.
17
Page 18
Proof of Lemma 8. Note that the random variables ϕτnpXi ´ qτnppq; pq, 1 ď i ď n are clearly centered because
qτnppq “ arg minuPR
EpητnpXi ´ u; pq ´ ητnpXi; pqq ñ EpϕτnpXi ´ qτnppq; pqq “ 0
by differentiating under the expectation sign. Write then
T1,n “ T1,1,n ` T1,2,n (B.10)
with T1,1,n “1
a
np1´ τnq
nÿ
i“1
1
rqτnppqsp´1
“
ϕτnpXi ´ qτnppq; pq1ltXiďqτn ppqu ´ E`
ϕτnpX ´ qτnppq; pq1ltXďqτn ppqu˘‰
and T1,2,n “1
a
np1´ τnq
nÿ
i“1
1
rqτnppqsp´1
“
ϕτnpXi ´ qτnppq; pq1ltXiąqτn ppqu ´ E`
ϕτnpX ´ qτnppq; pq1ltXąqτn ppqu˘‰
.
Here T1,1,n and T1,2,n are again sums of centered variables, which we analyse separately. The first term T1,1,n is
controlled by noting that since ρpnq ď 2a
φpnq (see Lemma 1.1 in Ibragimov, 1962), the seriesř8
n“1 ρpnq converges
and we may use Lemma 7 to get
VarpT1,1,nq “ O
˜
Var`
ϕτnpX ´ qτnppq; pq1ltXďqτn ppqu ´ E`
ϕτnpX ´ qτnppq; pq1ltXďqτn ppqu˘˘
p1´ τnqrqτnppqs2pp´1q
¸
.
Using Lemma 4(ii), we conclude that VarpT1,1,nq “ op1´ τnq, proving that
T1,1,nPÝÑ 0. (B.11)
We now work on T1,2,n. The essential step is to show that
T1,2,na
VarpT1,2,nq
dÝÑ N p0, 1q. (B.12)
For this, we use the Lindeberg-type central limit theorem of Utev (1990): taking, with the notation therein, jn “ 1
and kn “ n, and setting
T1,2,n “
nÿ
i“1
Vn,i
with Vn,i :“1
a
np1´ τnq
1
rqτnppqsp´1
“
ϕτnpXi ´ qτnppq; pq1ltXiąqτn ppqu ´ E`
ϕτnpX ´ qτnppq; pq1ltXąqτn ppqu˘‰
,
it is enough to show that
@ε ą 0,1
VarpT1,2,nq
nÿ
i“1
E´
V 2n,i1lt|Vn,i|ěε
?VarpT1,2,nqu
¯
Ñ 0 as nÑ8.
Because the Vn,i, 1 ď i ď n are identically distributed, by writing V 2n,i “ V 2`δ
n,i V´δn,i it is easy to see that this
convergence will be shown provided we prove that for some suitably small δ ą 0, the following Lyapunov condition
holds:nE|Vn,1|2`δ
rVarpT1,2,nqs1`δ{2Ñ 0 as nÑ8. (B.13)
To prove convergence (B.13), we first obtain an equivalent of the denominator. Apply Lemma 7 to get
Dc P r0,8q, limnÑ8
VarpT1,2,nq
nVarpVn,1q“ c.
18
Page 19
Note then that by strict stationarity,
VarpT1,2,nq
nVarpVn,1q“ 1` 2
nÿ
k“2
n´ k ` 1
ncorrpVn,1, Vn,kq.
It follows that c “ 1 in the case of independent observations; otherwise, the function x ÞÑ ϕτnpx´qτnppq; pq1ltxąqτn ppqu
is increasing, so that the positive quadrant dependence of pX1, Xkq implies that corrpVn,1, Vn,kq is nonnegative for
any k and n, see Lehmann (1966). Consequently
VarpT1,2,nq
nVarpVn,1q“ 1` 2
nÿ
k“2
n´ k ` 1
ncorrpVn,1, Vn,kq ě 1.
Letting nÑ8 shows that c ě 1 and therefore c “ 1` σ2 for some σ2 ě 0, as required. Besides, using Lemma 4(i)
entails
nVarpVn,1q Ñ 2γpp´ 1qBp2p´ 2, γ´1 ´ 2p` 2q
Bpp, γ´1 ´ p` 1qas nÑ8.
The formulas Bpx, yq “ ΓpxqΓpyq{Γpx` yq and Γpx` 1q “ xΓpxq now yield
2γpp´ 1qBp2p´ 2, γ´1 ´ 2p` 2q
Bpp, γ´1 ´ p` 1q“ V pγ; pq
so that
limnÑ8
VarpT1,2,nq “ p1` σ2q limnÑ8
nVarpVn,1q “ V pγ; pqp1` σ2q. (B.14)
Using this convergence, it follows that (B.13) and therefore convergence (B.12) will be shown if for some δ ą 0,
nE|Vn,1|2`δ Ñ 0. Choose now δ ą 0 so small that γ ă 1{rp2` δqpp´ 1qs and EpXp2`δqpp´1q´ q ă 8: the convergence
nE|Vn,1|2`δ Ñ 0 is then a straightforward consequence of the Holder inequality and Lemma 4(i). Hence (B.12),
which recalling (B.14) is exactly
T1,2,ndÝÑ N
`
0, V pγ; pqp1` σ2q˘
. (B.15)
Combine (B.10), (B.11) and (B.15) to conclude the proof.
Lemma 9. Work under the conditions of Theorem 1. Let
T2,npuq “ ´n
rqτnppqsp
ż uqτn ppq{?np1´τnq
0
rEpϕτnpX ´ qτnppq ´ t; pqq ´ EpϕτnpX ´ qτnppq; pqqsdt.
Then
T2,npuq Ñu2
2γas nÑ8.
Proof of Lemma 9. By Lemma 6, we obtain
EpϕτnpX ´ qτnppq ´ t; pqq ´ EpϕτnpX ´ qτnppq; pqq
“ p1´ 2τnqEp|X ´ qτnppq ´ t|p´1 signpX ´ qτnppq ´ tqp1ltXďqτn ppq`tu ´ 1ltXďqτn ppquqq
` E`
|τn ´ 1ltXďqτn ppqu|`
|X ´ qτnppq ´ t|p´1 signpX ´ qτnppq ´ tq ´ |X ´ qτnppq|
p´1 signpX ´ qτnppqq˘˘
,
19
Page 20
that is:
EpϕτnpX ´ qτnppq ´ t; pqq ´ EpϕτnpX ´ qτnppq; pqq
“ p1´ 2τnqEp|X ´ qτnppq ´ t|p´1 signpX ´ qτnppq ´ tqp1ltXąqτn ppqu ´ 1ltXąqτn ppq`tuqq
` τnE``
|X ´ qτnppq ´ t|p´1 signpX ´ qτnppq ´ tq ´ |X ´ qτnppq|
p´1 signpX ´ qτnppqq˘
1ltXąqτn ppqu˘
` p1´ τnqE``
|X ´ qτnppq ´ t|p´1 signpX ´ qτnppq ´ tq ´ |X ´ qτnppq|
p´1 signpX ´ qτnppqq˘
1ltXďqτn ppqu˘
“: p1´ 2τnqT2,1,nptq ` τnT2,2,nptq ` p1´ τnqT2,3,nptq.
The idea is now to control the three terms appearing in the above representation. In all these terms, |t| varies in
the interval Inpuq “ r0, |u|qτnppq{a
np1´ τnqs which is such that
sup|t|PInpuq
|t|
qτnppqÑ 0 as nÑ8. (B.16)
In this proof, all op¨q and Op¨q terms are to be understood as uniform in |t| P Inpuq. We also let Gpxq “ |x|p´1 signpxq,
whose (Lebesgue) derivative is gpxq “ pp´ 1q|x|p´2 on R.
First term T2,1,nptq: Writing
1ltXąqτn ppqu ´ 1ltXąqτn ppq`tu “
$
’
&
’
%
1ltqτn ppqăXďqτn ppq`tu if t ą 0
´1ltqτn ppq`tăXďqτn ppqu if t ă 0
it follows that:
T2,1,nptq “
$
’
&
’
%
EpGpX ´ qτnppq ´ tq1ltqτn ppqăXďqτn ppq`tuq if t ą 0
´EpGpX ´ qτnppq ´ tq1ltqτn ppq`tăXďqτn ppquq if t ă 0
“
$
’
&
’
%
EprGp´tq `şX
qτn ppqgpv ´ qτnppq ´ tqdvs1ltqτn ppqăXďqτn ppq`tuq if t ą 0
´EpşX
qτn ppq`tgpv ´ qτnppq ´ tqdv1ltqτn ppq`tăXďqτn ppquq if t ă 0
“
$
’
&
’
%
Gp´tqPpqτnppq ă X ď qτnppq ` tq `şqτn ppq`t
qτn ppqgpv ´ qτnppq ´ tqPpv ă X ď qτnppq ` tqdv if t ą 0
´şqτn ppq
qτn ppq`tgpv ´ qτnppq ´ tqPpv ă X ď qτnppqqdv if t ă 0
If condition H1pγq holds, then by (B.16) and Lemma 5(i) we get when t ą 0:
ż qτn ppq`t
qτn ppq
gpv ´ qτnppq ´ tqPpv ă X ď qτnppq ` tqdv
“ pp´ 1q
ż qτn ppq`t
qτn ppq
pqτnppq ` t´ vqp´2F pvq
qτnppq ` t´ v
γvp1` op1qqdv
“ pp´ 1qF pqτnppqq
γqτnppq
ż qτn ppq`t
qτn ppq
pqτnppq ` t´ vqp´1p1` op1qqdv
“p´ 1
ptpF pqτnppqq
γqτnppqp1` op1qq.
20
Page 21
If now we work under condition C2pγ, ρ,Aq, we can use Lemma 5(ii) instead to obtain
ż qτn ppq`t
qτn ppq
gpv ´ qτnppq ´ tqPpv ă X ď qτnppq ` tqdv
“ pp´ 1q
ż qτn ppq`t
qτn ppq
pqτnppq ` t´ vqp´2F pvq
„
qτnppq ` t´ v
γvp1` op1qq ` o
ˆ
A
ˆ
1
F pvq
˙˙
dv
“p´ 1
ptpF pqτnppqq
γqτnppqp1` op1qq ` o
˜
F pqτnppqqA
ˆ
1
F pqτnppqq
˙ż qτn ppq`t
qτn ppq
pqτnppq ` t´ vqp´2dv
¸
“p´ 1
ptpF pqτnppqq
γqτnppqp1` op1qq ` o
ˆ
F pqτnppqqA
ˆ
1
F pqτnppqq
˙
tp´1
˙
.
Likewise, when t ă 0 we have under condition H1pγq that:
´
ż qτn ppq
qτn ppq`t
gpv ´ qτnppq ´ tqPpv ă X ď qτnppqqdv “ ´1
pp´tqp
F pqτnppqq
γqτnppqp1` op1qq,
and under condition C2pγ, ρ,Aq that
´
ż qτn ppq
qτn ppq`t
gpv ´ qτnppq ´ tqPpv ă X ď qτnppqqdv “ ´1
pp´tqp
F pqτnppqq
γqτnppqp1` op1qq
` o
ˆ
F pqτnppqqA
ˆ
1
F pqτnppqq
˙
p´tqp´1
˙
.
Using Lemma 5(i) again and Proposition 1 we get, under condition H1pγq:
p1´ 2τnqT2,1,nptq “1
p|t|p
F pqτnppqq
γqτnppqp1` op1qq “ o
`
|t|rqτnppqsp´2F pqτnppqq
˘
“ o`
|t|rqτnppqsp´2p1´ τnq
˘
. (B.17)
Similarly, under condition C2pγ, ρ,Aq, we have by Lemma 5(ii) that
p1´ 2τnqT2,1,nptq “1
p|t|p
F pqτnppqq
γqτnppqp1` op1qq ` o
ˆ
F pqτnppqqA
ˆ
1
F pqτnppqq
˙
|t|p´1
˙
“ o`
|t|rqτnppqsp´2p1´ τnq
˘
` o´
rqτnppqsp´1n´1{2
?1´ τn
¯
. (B.18)
Second term T2,2,nptq: In the same spirit, write
T2,2,nptq
“ E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltXąqτn ppqudv
¸
“
$
’
&
’
%
E`ş
R gpvq1ltX´qτn ppqăvăX´qτn ppq´t,Xąqτn ppqudv˘
if t ă 0
´E`ş
R gpvq1ltX´qτn ppq´tăvăX´qτn ppq, Xąqτn ppqudv˘
if t ą 0
“
$
’
&
’
%
ş8
0gpvqPpqτnppq `maxp0, v ` tq ă X ă qτnppq ` vqdv if t ă 0
´ş8
´tgpvqPpqτnppq `maxp0, vq ă X ă qτnppq ` v ` tqdv if t ą 0
“
$
’
&
’
%
ş´t
0gpvqPpqτnppq ă X ă qτnppq ` vqdv `
ş8
´tgpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv if t ă 0
´ş0
´tgpvqPpqτnppq ă X ă qτnppq ` v ` tqdv ´
ş8
0gpvqPpqτnppq ` v ă X ă qτnppq ` v ` tqdv if t ą 0.
21
Page 22
When t ă 0, we get by (B.16) and Lemma 5(i):
ż ´t
0
gpvqPpqτnppq ă X ă qτnppq ` vqdv “
ż ´t
0
gpvqF pqτnppqqv
γqτnppqp1` op1qqdv
“F pqτnppqq
γqτnppqpp´ 1q
ż ´t
0
vp´1p1` op1qqdv
“p´tqp
ppp´ 1q
F pqτnppqq
γqτnppqp1` op1qq
when H1pγq holds. Working under C2pγ, ρ,Aq instead and using Lemma 5(ii) entails
ż ´t
0
gpvqPpqτnppq ă X ă qτnppq ` vqdv
“
ż ´t
0
gpvqF pqτnppqq
„
v
γqτnppqp1` op1qq ` o
ˆ
A
ˆ
1
F pqτnppqq
˙˙
dv
“p´tqp
ppp´ 1q
F pqτnppqq
γqτnppqp1` op1qq ` o
ˆ
F pqτnppqqA
ˆ
1
F pqτnppqq
˙
p´tqp´1
˙
.
Furthermore, applying Lemma 5 again yields:
ż 8
´t
gpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv “
ż 8
´t
gpvqF pqτnppq ` v ` tq´t
γpqτnppq ` v ` tqp1` op1qqdv
under H1pγq, and
ż 8
´t
gpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv
“
ż 8
´t
gpvqF pqτnppq ` v ` tq
„
´t
γpqτnppq ` v ` tqp1` op1qq ` o
ˆ
A
ˆ
1
F pqτnppq ` v ` tq
˙˙
dv
under C2pγ, ρ,Aq. Let ε ą 0 be such that 2 ` γ´1 ´ p ´ ε ą 0. By a uniform convergence theorem for regularly
varying functions (see Theorem 1.5.2 in Bingham et al., 1987) we obtain
supxě1
ˇ
ˇ
ˇ
ˇ
pqxq´ε`1{γF pqxq
q´ε`1{γF pqq´ x´ε
ˇ
ˇ
ˇ
ˇ
Ñ 0 as q Ñ `8.
As a consequence
ż 8
´t
gpvqF pqτnppq ` v ` tq´t
γpqτnppq ` v ` tqp1` op1qqdv
“´t
γpp´ 1qrqτnppqs
1{γF pqτnppqq
ż 8
´t
vp´2pqτnppq ` v ` tq´1´1{γdv
` o
ˆ
´trqτnppqs1{γ´εF pqτnppqq
ż 8
´t
vp´2pqτnppq ` v ` tq´1´1{γ`εdv
˙
.
Now, for n large enough and all w ě 0,
0 ď wp´2
ˆ
1` w `t
qτnppq
˙´1´1{γ
ď wp´2
ˆ
1
2` w
˙´1´1{γ
where the right-hand side defines an integrable function on p0,8q. By the dominated convergence theorem, we get
ż 8
´t
vp´2pqτnppq ` v ` tq´1´1{γdv “ rqτnppqs
p´2´1{γ
ż 8
´t{qτn ppq
wp´2
ˆ
1` w `t
qτnppq
˙´1´1{γ
dw
“ rqτnppqsp´2´1{γ
ż 8
0
wp´2 p1` wq´1´1{γ
dwp1` op1qq.
22
Page 23
The change of variables z “ p1` wq´1 yields
ż 8
0
wp´2 p1` wq´1´1{γ
dw “
ż 1
0
p1´ zqp´2z1`1{γ´pdz “ Bpp´ 1, 2` γ´1 ´ pq.
Similarly
ż 8
´t
vp´2pqτnppq ` v ` tq´1´1{γ`εdv “ rqτnppqs
p´2´1{γ`εBpp´ 1, 2` γ´1 ´ p´ εqp1` op1qq
so that under H1pγq:
ż 8
´t
gpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv “´t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq.
When C2pγ, ρ,Aq holds, because the function F ˆA ˝ p1{F q is regularly varying with index pρ´ 1q{γ ď ´1{γ and
therefore
p´ 2`ρ´ 1
γď p´ 2´
1
γă p´ 2` p2´ 2pq “ ´p ă ´1,
we can argue along the same lines to obtain
ż 8
´t
gpvqF pqτnppq ` v ` tqA
ˆ
1
F pqτnppq ` v ` tq
˙
dv
“ pp´ 1qrqτnppqsp1´ρq{γF pqτnppqqA
ˆ
1
F pqτnppqq
˙ż 8
´t
vp´2pqτnppq ` v ` tqpρ´1q{γdvp1` op1qq
“ O
ˆ
rqτnppqsp´1F pqτnppqqA
ˆ
1
F pqτnppqq
˙˙
.
Thus
ż 8
´t
gpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv “´t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq
` o
ˆ
rqτnppqsp´1F pqτnppqqA
ˆ
1
F pqτnppqq
˙˙
.
When t ą 0 and H1pγq holds, we get in a similar fashion
ż 0
´t
gpvqPpqτnppq ă X ă qτnppq ` v ` tqdv “tp
p
F pqτnppqq
γqτnppqp1` op1qq
and
ż 8
0
gpvqPpqτnppq ` v ă X ă qτnppq ` v ` tqdv “t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq.
If C2pγ, ρ,Aq holds, we have
ż 0
´t
gpvqPpqτnppq ă X ă qτnppq ` v ` tqdv “tp
p
F pqτnppqq
γqτnppqp1` op1qq ` o
ˆ
F pqτnppqqA
ˆ
1
F pqτnppqq
˙
tp´1
˙
and
ż 8
0
gpvqPpqτnppq ` v ă X ă qτnppq ` v ` tqdv “t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq
` o
ˆ
rqτnppqsp´1F pqτnppqqA
ˆ
1
F pqτnppqq
˙˙
.
23
Page 24
All in all, under H1pγq, using (B.16) entails:
T2,2,nptq
“
$
’
’
’
&
’
’
’
%
´t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq `p´tqp
ppp´ 1q
F pqτnppqq
γqτnppqp1` op1qq if t ă 0
´t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq ´tp
p
F pqτnppqq
γqτnppqp1` op1qq if t ą 0
“ ´t
γpp´ 1qrqτnppqs
p´2F pqτnppqqBpp´ 1, 2` γ´1 ´ pqp1` op1qq.
Working under C2pγ, ρ,Aq gives instead:
T2,2,nptq “ ´t
γpp´1qrqτnppqs
p´2F pqτnppqqBpp´1, 2`γ´1´pqp1`op1qq`o
ˆ
rqτnppqsp´1F pqτnppqqA
ˆ
1
F pqτnppqq
˙˙
because of (B.16) again. By Proposition 1 and the identity
@x, y ą 0,Bpx, y ` 1q
Bpx` 1, yq“
Γpxq
Γpx` 1q
Γpy ` 1q
Γpyq“y
x
this reads
τnT2,2,nptq “ ´tpγ´1 ´ pp´ 1qqrqτnppqs
p´2p1´ τnqp1` op1qq (B.19)
under H1pγq, and
τnT2,2,nptq “ ´tpγ´1 ´ pp´ 1qqrqτnppqs
p´2p1´ τnqp1` op1qq ` o´
rqτnppqsp´1n´1{2
?1´ τn
¯
(B.20)
under C2pγ, ρ,Aq.
Third term T2,3,nptq: Write
T2,3,nptq “ E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltXďqτn ppqudv
¸
.
Split then the above integral as
E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltXďqτn ppqudv
¸
“ E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltXďqτn ppq{2udv
¸
` E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltqτn ppq{2ăXďqτn ppqudv
¸
.
The first term in the rhs above is
pp´ 1qE
˜
|X ´ qτnppq|p´2rX ´ qτnppqs
ż 1´t{pX´qτn ppqq
1
|w|p´2dw1ltX´qτn ppqď´qτn ppq{2u
¸
.
Because sup|t|PInpuq |t|{qτnppq Ñ 0 and |X ´ qτnppq| ě qτnppq{2 in the integrand, this term is equivalent to
´tpp´ 1qE`
|X ´ qτnppq|p´21ltX´qτn ppqď´qτn ppq{2u
˘
“ ´tpp´ 1qrqτnppqsp´2E
˜
ˇ
ˇ
ˇ
ˇ
X
qτnppq´ 1
ˇ
ˇ
ˇ
ˇ
p´2
1ltXďqτn ppq{2u
¸
“ ´tpp´ 1qrqτnppqsp´2p1` op1qq
24
Page 25
by the dominated convergence theorem. The second term, meanwhile, is equal to
E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltqτn ppq{2ăXďqτn ppqudv
¸
“
$
’
&
’
%
E`ş
R gpvq1ltX´qτn ppqăvăX´qτn ppq´t, qτn ppq{2ăXďqτn ppqudv˘
if t ă 0
´E`ş
R gpvq1ltX´qτn ppq´tăvăX´qτn ppq, qτn ppq{2ăXďqτn ppqudv˘
if t ą 0
“
$
’
&
’
%
ş
R gpvqPpqτnppq `maxp´qτnppq{2, v ` tq ă X ă qτnppq `minp0, vqqdv if t ă 0
´ş
R gpvqPpqτnppq `maxp´qτnppq{2, vq ă X ă qτnppq `minp0, v ` tqqdv if t ą 0.
When H1pγq holds, we then have by Lemma 5(i):ˇ
ˇ
ˇ
ˇ
ˇ
E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltqτn ppq{2ăXďqτn ppqudv
¸ˇ
ˇ
ˇ
ˇ
ˇ
“
$
’
&
’
%
ş´t
´qτn ppq{2gpvqPpqτnppq `maxp´qτnppq{2, v ` tq ă X ă qτnppq `minp0, vqqdv if t ă 0
ş0
´t´qτn ppq{2gpvqPpqτnppq `maxp´qτnppq{2, vq ă X ă qτnppq `minp0, v ` tqqdv if t ą 0
ď
$
’
&
’
%
ş´t
´qτn ppq{2gpvqPpqτnppq ` v ` t ă X ă qτnppq ` vqdv if t ă 0
ş0
´t´qτn ppq{2gpvqPpqτnppq ` v ă X ă qτnppq ` v ` tqdv if t ą 0
“
$
’
’
’
&
’
’
’
%
ş´t
´qτn ppq{2gpvqF pqτnppq ` v ` tq
´t
γpqτnppq ` v ` tqdvp1` op1qq if t ă 0
ş0
´t´qτn ppq{2gpvqF pqτnppq ` vq
t
γpqτnppq ` vqdvp1` op1qq if t ą 0
ď F ppqτnppq{2q ´ |t|q|t|
γppqτnppq{2q ´ |t|q
$
’
&
’
%
ş´t
´qτn ppq{2gpvqdvp1` op1qq if t ă 0
ş0
´t´qτn ppq{2gpvqdvp1` op1qq if t ą 0
“ F pqτnppq{2q|t|
γ
ˆ
qτnppq
2
˙p´2
p1` op1qq “ o`
|t|rqτnppqsp´2
˘
.
Similarly, when C2pγ, ρ,Aq holds, we have by Lemma 5(ii):ˇ
ˇ
ˇ
ˇ
ˇ
E
˜
ż X´qτn ppq´t
X´qτn ppq
gpvq1ltqτn ppq{2ăXďqτn ppqudv
¸ˇ
ˇ
ˇ
ˇ
ˇ
ď
$
’
’
’
&
’
’
’
%
ş´t
´qτn ppq{2gpvqF pqτnppq ` v ` tq
„
´t
γpqτnppq ` v ` tqp1` op1qq ` o
ˆ
A
ˆ
1
F pqτnppq ` v ` tq
˙˙
dv if t ă 0
ş0
´t´qτn ppq{2gpvqF pqτnppq ` vq
„
t
γpqτnppq ` vqp1` op1qq ` o
ˆ
A
ˆ
1
F pqτnppq ` vq
˙˙
dv if t ą 0
“ O`
|t|F pqτnppqqrqτnppqsp´2
˘
` o
ˆ
F pqτnppqqrqτnppqsp´1A
ˆ
1
F pqτnppqq
˙˙
“ o`
|t|rqτnppqsp´2
˘
` o´
rqτnppqsp´1n´1{2
?1´ τn
¯
.
As a conclusion, if H1pγq holds:
p1´ τnqT2,3,nptq “ ´tpp´ 1qrqτnppqsp´2p1´ τnqp1` op1qq (B.21)
and if C2pγ, ρ,Aq holds then:
p1´ τnqT2,3,nptq “ p1´ τnq”
´tpp´ 1qrqτnppqsp´2p1` op1qq ` o
´
rqτnppqsp´1n´1{2
?1´ τn
¯ı
. (B.22)
25
Page 26
Combining (B.17), (B.19) and (B.21), we get
p1´ 2τnqT2,1,nptq ` τnT2,2,nptq ` p1´ τnqT2,3,nptq “ ´t
γrqτnppqs
p´2p1´ τnqp1` op1qq
and thus
T2,npuq “ ´n
rqτnppqsp
ż uqτn ppq{?np1´τnq
0
rEpϕτnpX ´ qτnppq ´ t; pqq ´ EpϕτnpX ´ qτnppq; pqqsdt
“np1´ τnq
γrqτnppqs2
ż uqτn ppq{?np1´τnq
0
t dtp1` op1qq
“u2
2γp1` op1qq
if H1pγq is assumed. If we work under C2pγ, ρ,Aq, we have by combining (B.18), (B.20) and (B.22) that:
p1´ 2τnqT2,1,nptq ` τnT2,2,nptq ` p1´ τnqT2,3,nptq “ ´t
γrqτnppqs
p´2p1´ τnqp1` op1qq
` o´
rqτnppqsp´1n´1{2
?1´ τn
¯
and therefore
T2,npuq “ ´n
rqτnppqsp
ż uqτn ppq{?np1´τnq
0
rEpϕτnpX ´ qτnppq ´ t; pqq ´ EpϕτnpX ´ qτnppq; pqqsdt
“n
rqτnppqsp
«
1
γrqτnppqs
p´2p1´ τnq
ż uqτn ppq{?np1´τnq
0
t dtp1` op1qq ` o
ˆ
rqτnppqsp
n
˙
ff
“u2
2γp1` op1qq.
The proof is complete.
Lemma 10. Work under the conditions of Theorem 1. Let Sn,ipvq :“ ϕτnpXi ´ v; pq ´ EpϕτnpX ´ v; pqq and
T3,npuq “ ´1
rqτnppqsp
nÿ
i“1
ż uqτn ppq{?np1´τnq
0
rSn,ipqτnppq ` tq ´ Sn,ipqτnppqqsdt.
Then
T3,npuqPÝÑ 0 as nÑ8.
Proof of Lemma 10. As in the proof of Lemma 9, let Inpuq “ r0, |u|qτnppq{a
np1´ τnqs. In the present proof, all
op¨q and Op¨q terms are to be understood as uniform in |t| P Inpuq.
Let Snpvq :“ ϕτnpX ´ v; pq ´ EpϕτnpX ´ v; pqq. By Lemma 7,
VarpT3,npuqq “ O
˜
n
rqτnppqs2p
Var
˜
ż uqτn ppq{?np1´τnq
0
rSnpqτnppq ` tq ´ Snpqτnppqqsdt
¸¸
.
Because for any t, Snpqτnppq ` tq is centered and rϕτnpX ´ qτnppq ´ t; pqs2 is integrable w.r.t. t on the interval
r0, uqτnppq{a
np1´ τnqs, we get
VarpT3,npuqq
“ O
˜
n
rqτnppqs2p
ż
r0, uqτn ppq{?np1´τnqs2
EprSnpqτnppq ` sq ´ SnpqτnppqqsrSnpqτnppq ` tq ´ Snpqτnppqqsqds dt
¸
.
26
Page 27
The Cauchy-Schwarz inequality now yields
VarpT3,npuqq ďn
rqτnppqs2p
˜
ż uqτn ppq{?np1´τnq
0
a
Ep|Snpqτnppq ` tq ´ Snpqτnppqq|2q dt
¸2
. (B.23)
Applying Lemma 6, we get for any t
|Snpqτnppq ` tq ´ Snpqτnppqq|
ď |ϕτnpX ´ qτnppq ´ t; pq ´ ϕτnpX ´ qτnppq; pq| ` E|ϕτnpX ´ qτnppq ´ t; pq ´ ϕτnpX ´ qτnppq; pq|
ď |t|p´1`
1lt|X´qτn ppq|ď|t|u ` Pp|X ´ qτnppq| ď |t|q˘
` p1´ τn ` 1ltXąqτn ppquq
$
’
’
’
&
’
’
’
%
2|t|p´1 if 1 ă p ă 2
pp´ 1qp2p´2 ` 1qp|t|p´1 ` |X ´ qτnppq|p´2|t|q if p ě 2
` p1´ τnq
$
’
’
’
&
’
’
’
%
2|t|p´1 if 1 ă p ă 2
pp´ 1qp2p´2 ` 1qp|t|p´1 ` |t|E|X ´ qτnppq|p´2q if p ě 2
`
$
’
’
’
&
’
’
’
%
2|t|p´1PpX ą qτnppqq if 1 ă p ă 2
pp´ 1qp2p´2 ` 1qp|t|p´1PpX ą qτnppqq ` |t|Ep|X ´ qτnppq|p´21ltXąqτn ppquqq if p ě 2.
By Lemma 1 with Hpxq “ px´ 1qp´21ltxě1u and Proposition 1, we have when p ě 2 that
Ep|X ´ qτnppq|p´21ltXąqτn ppquq “ rqτnppqsp´2E
˜
„
X
qτnppq´ 1
p´2
1ltXąqτn ppqu
¸
“ O`
p1´ τnqrqτnppqsp´2
˘
and it is moreover a consequence of the dominated convergence theorem that
Ep|X ´ qτnppq|p´2q “ rqτnppqsp´2E
˜
ˇ
ˇ
ˇ
ˇ
X
qτnppq´ 1
ˇ
ˇ
ˇ
ˇ
p´2¸
“ rqτnppqsp´2p1` op1qq.
Recalling convergence (B.16), i.e. |t|{qτnppq Ñ 0 uniformly in |t| P Inpuq, and using Proposition 1 again, we get
|Snpqτnppq ` tq ´ Snpqτnppqq|
ď |t|p´1`
1lt|X´qτn ppq|ď|t|u ` Pp|X ´ qτnppq| ď |t|q˘
` p1´ τn ` 1ltXąqτn ppquq
$
’
’
’
&
’
’
’
%
2|t|p´1 if 1 ă p ă 2
pp´ 1qp2p´2 ` 1qp|t|p´1 ` |X ´ qτnppq|p´2|t|q if p ě 2
`
$
’
’
’
&
’
’
’
%
Opp1´ τnq|t|p´1q if 1 ă p ă 2
O`
p1´ τnqrqτnppqsp´2|t|
˘
if p ě 2.
27
Page 28
Squaring, integrating and using convergence (B.16) once again, we obtain that there is a constant C with
E|Snpqτnppq ` tq ´ Snpqτnppqq|2 ď C|t|2pp´1qPp|X ´ qτnppq| ď |t|q
`
$
’
’
’
&
’
’
’
%
Opp1´ τnq|t|2pp´1qq if 1 ă p ă 2
O`
p1´ τnqrqτnppqs2pp´2q|t|2
˘
if p ě 2.
When H1pγq holds, then by Lemma 5(i) and Proposition 1 again,
Pp|X ´ qτnppq| ď |t|q “ F pqτnppqq2|t|
γqτnppqp1` op1qq “ op1´ τnq.
If we work under C2pγ, ρ,Aq, then by Lemma 5(ii) and Proposition 1,
Pp|X ´ qτnppq| ď |t|q “ F pqτnppqq
„
2|t|
γqτnppqp1` op1qq ` o
ˆ
A
ˆ
1
F pqτnppqq
˙˙
“ op1´ τnq.
In any case,
E|Snpqτnppq ` tq ´ Snpqτnppqq|2 ď
$
’
’
’
&
’
’
’
%
Opp1´ τnq|t|2pp´1qq if 1 ă p ă 2
O`
p1´ τnqrqτnppqs2pp´2q|t|2
˘
`O`
p1´ τnq|t|2pp´1q
˘
if p ě 2
“
$
’
’
’
&
’
’
’
%
Opp1´ τnq|t|2pp´1qq if 1 ă p ă 2
O`
p1´ τnqrqτnppqs2pp´2q|t|2
˘
if p ě 2
by using (B.16). Because
n
rqτnppqs2p
˜
ż uqτn ppq{?np1´τnq
0
b
p1´ τnq|t|2pp´1q dt
¸2
“np1´ τnq
rqτnppqs2p
˜
ż uqτn ppq{?np1´τnq
0
|t|p´1 dt
¸2
“ Oprnp1´ τnqs1´pq “ op1q
and
n
rqτnppqs2p
˜
ż uqτn ppq{?np1´τnq
0
b
p1´ τnqrqτnppqs2pp´2q|t|2 dt
¸2
“np1´ τnq
rqτnppqs4
˜
ż uqτn ppq{?np1´τnq
0
|t| dt
¸2
“ Oprnp1´ τnqs´1q “ op1q
we get T3,npuqPÝÑ 0 and the proof is complete.
C Additional simulations
C.1 Extreme expectile estimation
We concentrate here on extreme L2´quantiles, or equivalently, expectiles. A comparison of the three estimators
pqWαnp2q in (11), rqWαnp2q in (12) and qqpαnp2q in (13) (see the main paper) of the extreme expectile qαnp2q is shown in
Figures 1 and 2, where we present the evolution of their relative MSE (in log scale) in terms of the value k. We used
28
Page 29
the same considerations as in Section 6 for the choice of pγn and the intermediate and extreme expectile levels τn and
τ 1n “ αn. The experiments employ the Frechet, Pareto and Student distributions with tail-indices γ P t0.1, 0.45u
and various values of p P p1, 2q in the formulation (13) of qqpαnp2q.
In the case of Frechet and Pareto distributions, we already know that rqWαnp2q behaves better than pqWαnp2q in terms
of relative MSE. In this case, it turns out that the accuracy of the estimator qqpαnp2q is also superior to pqWαnp2q and
is similar to that of rqWαnp2q for very ‘small’ values of p (close to 1), as may be seen from Figure 1. In this Figure we
only present the estimates of the relative MSE in a log scale. We do not graph the bias estimates to save space:
most of the error is due to variance, the squared bias being much smaller in all cases.
In contrast, in the case of the Student distribution, we know that pqWαnp2q behaves overall better than rqWαnp2q:
we do not graph the curve related to the latter estimator in Figure 2 as it exhibits considerable volatility. Also, it
appears in this case that qqpαnp2q outperforms rqWαnp2q as well and has a similar behavior compared to to pqWαnp2q for
very ‘large’ values of p (close to 2), as may be seen from Figure 2. There is also a significant improvement for large
γ in this case when using the estimator qqpαnp2q, probably because this estimator benefits from increasing robustness
(see the final lines of Section 6.2).
This might suggest the following strategy with a real data set. If the data set is concerned with a non-negative
loss distribution, it is most efficient to use rqWαnp2q and qqpαnp2q with values of p very close to 1. At the opposite,
if the data set is concerned with a real-valued profit-loss random variable, we favor the use of pqWαnp2q and qqpαnp2q
with values of p very close to 2. The important question of how to pick out p in practice, in order to get the best
estimates qqpαnp2q from historical data, can be addressed by adapting the practical guidelines provided in Section 7
for selecting p in the extreme Lp´quantile estimates rqWαnppq and pqWαnppq.
C.2 Extreme expectile composite estimation
Here, we focus on the composite Lp´estimator pqWpτ 1npp,αn;2q
ppq in (16) of the extreme expectile qαnp2q, where αn “
1 ´ 1{n. A comparison with the benchmark estimators pqWαnp2q in (11) and rqWαnp2q in (12) is shown below in
Figures 3 and 4. We used the same considerations as in Section 6 and Supplement C.1 for the choice of pγn and
the intermediate and extreme expectile levels τn and τ 1n ” αn. All the experiments employ the Frechet, Pareto and
Student distributions with tail-indices γ P t0.1, 0.45u and various values of the power p P p1, 2q in the formulation
of pqWpτ 1npp,αn;2q
ppq.
In Frechet and Pareto models, where rqWαnp2q is known to be superior to pqWαnp2q in terms of MSE, it may be seen
from Figure 3 that pqWpτ 1npp,αn;2q
ppq behaves similarly to rqWαnp2q for very small values of p (close to 1). Our simulations
also indicate that most of the error is due to variance, the squared bias being much smaller in all cases. We only
display in Figure 3 the estimates of the relative MSE (in a log scale) to save space.
In the Student model, where pqWαnp2q is known to be superior to rqWαnp2q, it may be seen from Figure 4 that
pqWpτ 1npp,αn;2q
ppq performs at least like pqWαnp2q, for large values of p (close to 2), in terms of both MSE (top panels) and
Bias (bottom panels). Interestingly, like qqpαnp2q in Supplement C.1, the composite estimator pqWpτ 1npp,αn;2q
ppq seems to
provide a better accuracy relative to pqWαnp2q in the case of heavier tails pγ “ 0.45q.
As regards the second composite estimator rqWpτ 1npp,αn;2q
ppq in (17), the obtained Monte Carlo estimates do not
provide evidence of any added value with respect to the benchmark estimators pqWαnp2q and rqWαnp2q, hence the results
29
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0.025
0.050
0.075
0.100
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Fréchet : gamma=0.1
0.12
0.14
0.16
0.18
0 50 100 150k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Fréchet : gamma=0.45
0.060
0.064
0.068
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Pareto : gamma=0.1
0.08
0.10
0.12
0.14
0.16
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Pareto : gamma=0.45
Figure 1: Frechet and Pareto distributions—RMSE (in log scale) of pqWαnp2q (blue), rqWαnp2q (red), and qqpαnp2q with
p “ 1.001 (black), p “ 1.0005 (grey), p “ 1.0001 (yellow) and p “ 1.00005 (green). From left to right, γ “ 0.1, 0.45.
From top to bottom, Frechet and Pareto distributions.
are not reported here.
C.3 Quality of asymptotic approximations
This section gives Monte Carlo evidence that our limit theorems provide adequate approximations for finite sample
sizes. We first investigate the normality of the extrapolated least asymmetrically weighted Lp estimators pqWτ 1n ppq
in (6) and the plug-in Weissman estimators rqWτ 1n ppq in (7), for τ 1n “ 1 ´ 1{n and p P t1.2, 1.5, 1.8u. Hereafter we
restrict our simulation study to the Student distribution with independent observations. The asymptotic normality
of pqWτ 1n ppq{qτ1nppq in Theorem 2 can be expressed as rn logppqWτ 1n ppq{qτ
1nppqq
dÝÑ ζ, with rn “
?k
logrk{pnp1´τ 1nqqs. Likewise,
the asymptotic normality of rqWτ 1n ppq{qτ1nppq in Theorem 3 can be expressed as rn logprqWτ 1n ppq{qτ
1nppqq
dÝÑ ζ. Following
Theorems 2.4.1 and 3.2.5 in de Haan and Ferreira (2006, p.50 and p.74), the limit distribution ζ of the Hill estimator
under independence is N pλ{p1´ρq, γ2q, where λ “ limnÑ8
?kApnk q. It can be shown that a Student tν distribution
satisfies the conditions of the two aforementioned theorems with γ “ 1{ν, ρ “ ´2{ν and
Aptq „ν ` 1
ν ` 2pcνtq
´2{ν , cν “2Γppν ` 1q{2qνpν´1q{2
?νπΓpν{2q
.
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0.07
0.09
0.11
0.13
10 20 30 40k
RMSE
(in lo
g sca
le)
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.1
0.12
0.14
0.16
0.18
10 20 30 40k
RMSE
(in lo
g sca
le)
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.45
0.0
0.2
0.4
0.6
10 20 30 40k
Bias
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.1
0.0
0.2
0.4
0.6
0.8
10 20 30 40k
Bias
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.45
Figure 2: Student distribution. Top—RMSE (in log scale) of pqWαnp2q (blue) and qqpαnp2q with p “ 1.999 (black),
p “ 1.9995 (grey), p “ 1.9999 (yellow), p “ 1.99995 (green) and p “ 1.99999 (pink). From left to right, γ “ 0.1, 0.45.
Bottom—Bias estimates.
Hence, we can compare the distributions of
xWn :“”
rn logppqWτ 1n ppq{qτ 1nppqq ´ λ{p1´ ρqı
{γ and ĂWn :“”
rn logprqWτ 1n ppq{qτ 1nppqq ´ λ{p1´ ρqı
{γ
with the limit distribution N p0, 1q, with λ «?kApnk q for n large enough. The Q–Q-plots in Figures 5 and 6 present,
respectively, the sample quantiles of xWn and ĂWn, based on 3, 000 simulated samples of size n “ 1000, versus the
theoretical standard normal quantiles. For each estimator, we used the optimal k selected by the data-driven method
described in Section 6.3 of the main article. It may be seen that the scatters for the Student t1{γ distributions, with
γ “ 0.1, 0.45 displayed respectively from left to right, are quite encouraging for all values of p.
Next, we investigate the normality of the estimators qqpαnp2q in (13) and pqWpτ 1npp,αn;2q
ppq in (16) of the extreme
expectile qαnp2q, where αn “ 1´1{n. For the Student distribution we used large values of p (close to 2) as suggested
by our experiments in Sections C.1 and C.2, but also smaller values of p, namely p P t1.2, 1.5, 1.8, 1.99, 1.999, 1.9999u.
The asymptotic normality of qqpαnp2q{qαnp2q in Theorem 7 can be expressed as vn logpqqpαnp2q{qαnp2qqdÝÑ ζ, with
vn “?k
logrk{pnp1´αnqqs. Likewise, the asymptotic normality of pqW
pτ 1npp,αn;2qppq{qαnp2q in Theorem 8 can be expressed as
31
Page 32
0.025
0.050
0.075
0.100
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Fréchet : gamma=0.1
0.12
0.14
0.16
0.18
0 50 100 150k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Fréchet : gamma=0.45
0.060
0.064
0.068
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Pareto : gamma=0.1
0.08
0.10
0.12
0.14
0.16
0 50 100 150 200k
RMSE
(in lo
g sca
le)
variablehat
tilde
p=1.001
p=1.0005
p=1.0001
p=1.00005
Pareto : gamma=0.45
Figure 3: Frechet and Pareto distributions—RMSE (in log scale) of pqWαnp2q (blue), rqWαnp2q (red), and pqWpτ 1npp,αn;2q
ppq
with p “ 1.001 (black), p “ 1.0005 (grey), p “ 1.0001 (yellow) and p “ 1.00005 (green). From left to right,
γ “ 0.1, 0.45. From top to bottom, Frechet and Pareto distributions.
vn logppqWpτ 1npp,αn;2q
ppq{qαnp2qqdÝÑ ζ. Thus we can compare the distributions of
|Wn :““
vn logpqqpαnp2q{qαnp2qq ´ λ{p1´ ρq‰
{γ and xWpτ 1npp,αn;2q
:“”
vn logppqWpτ 1npp,αn;2q
ppq{qαnp2qq ´ λ{p1´ ρqı
{γ
with the limit distribution N p0, 1q. The Q–Q-plots in Figures 7 and 8 present, respectively, the sample quantiles of
|Wn and xWpτ 1npp,αn;2q
, based on 3, 000 simulated samples of size n “ 1000 as above, versus the theoretical standard
normal quantiles. For each estimator, we used the optimal k selected by the data-driven method. The scatters for
the Student t1{γ distributions, with γ “ 0.1, 0.45 respectively from left to right, indicate that the limit Theorems 7
and 8 also provide adequate approximations for finite sample sizes.
D Medical insurance data example
We consider here the Society of Actuaries’ Group Medical Insurance Large Claims Database which records all the
claim amounts exceeding 25,000 USD over the period 1991-92. Similarly to Beirlant et al. (2004), we focus on the
75,789 claims for 1991 that we treat as the outcomes of i.i.d. non-negative loss random variables X1, . . . , Xn. The
scatterplot and histogram of the log-claim amounts in Figure 9 (a) give evidence of an important right-skewness.
32
Page 33
0.09
0.11
0.13
10 20 30 40k
RMSE
(in lo
g sca
le)
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.1
0.12
0.14
0.16
0.18
10 20 30 40k
RMSE
(in lo
g sca
le)
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.45
0.0
0.2
0.4
0.6
10 20 30 40k
Bias
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.1
0.0
0.2
0.4
0.6
0.8
10 20 30 40k
Bias
variablehat
p=1.999
p=1.9995
p=1.9999
p=1.99995
p=1.99999
Student : gamma=0.45
Figure 4: Student distribution. Top—RMSE (in log scale) of pqWτ 1n p2q (blue) and pqWpτ 1npp,αn;2q
ppq with p “ 1.999 (black),
p “ 1.9995 (grey), p “ 1.9999 (yellow), p “ 1.99995 (green) and p “ 1.99999 (pink). From left to right, γ “ 0.1, 0.45.
Bottom—Bias estimates.
The model assumption of a heavy-tailed loss severity distribution has been already verified in Beirlant et al. (2004,
p.123) with Hill’s estimate pγn around 0.35. Insurance companies typically are interested in an estimate of the claim
amount that will be exceeded (on average) only once in 100,000 cases. This translates into estimating the extreme
quantile qαnp1q with the relative frequency αn “ 1´ 1100,000 ą 1´ 1
n , or equivalently, the generalized Lp´quantile
qτ 1nppq ” qαnp1q with the extreme level τ 1n :“ τ 1npp, αn; 1q described in (9). The Value at Risk qαnp1q ” qτ 1npp,αn;1qppq
can be estimated either by the traditional Weissman quantile estimator pqWαnp1q defined in (8), or by the composite
Lp´quantile estimator pqWpτ 1npp,αn;1q
ppq studied in Theorem 5. To calculate the two estimates pqWαnp1q and pqWpτ 1npp,αn;1q
ppq
of the VaR as well as the estimate pτ 1npp, αn; 1q of τ 1npp, αn; 1q defined in (10), we used the optimal sample fraction
k selected by the data-driven method described in Section 6.3. The final composite estimates pqWpτ 1npp,αn;1q
ppq are
plotted in Figure 9 (b) against the power p in blue, along with the constant traditional estimate pqWαnp1q in green
and the sample maximum in red. None of the two extrapolated VaR estimates pqWpτ 1npp,αn;1q
ppq and pqWαnp1q exceed
the sample maximum Xn,n “ 4, 518, 420 USD. The classical L1´quantile based estimator pqWαnp1q relies on a single
order statistic pqτnp1q, and hence may not respond properly to infrequent large claims. By contrast, the composite
Lp´quantile estimator pqWpτ 1npp,αn;1q
ppq relies directly on the least asymmetrically weighted Lp estimator pqτnppq given
33
Page 34
Figure 5: Q–Q-plots on quality of asymptotic approximations. Each plot shows the sample quantiles of xWn versus
the theoretical standard normal quantiles, based on 3, 000 samples of size n “ 1000. Data are simulated from the
Student t1{γ with γ “ 0.1 (left panels) and γ “ 0.45 (right panels). From top to bottom, p “ 1.2, 1.5, 1.8.
Figure 6: As before with scatters for ĂWn.
34
Page 35
Figure 7: Q–Q-plots on quality of asymptotic approximations. Each plot shows the sample quantiles of |Wn versus the
theoretical standard normal quantiles, based on 3, 000 samples of size n “ 1000. Data are simulated from the Student
t1{γ with γ “ 0.1 (left panels) and γ “ 0.45 (right panels). From top to bottom, p “ 1.2, 1.5, 1.8, 1.99, 1.999, 1.9999.
35
Page 36
Figure 8: As before with scatters for xWpτ 1npp,αn;2q
.
36
Page 37
in (3), and hence it bears much better the burden of representing a conservative measure of risk. It may also be
seen from the path p ÞÑ pqWpτ 1npp,αn;1q
ppq that this risk measure tends to be more alert to infrequent large claims as the
power p increases.
The resulting estimates pτ 1npp, αn; 1q of the extreme level τ 1n :“ τ 1npp, αn; 1q such that qτ 1nppq ” qαnp1q are plotted
in Figure 9 (c) against p in blue, along with the constant tail probability αn in red horizontal line. This plot is of
course of capital importance when it comes to use a generalized Lp´quantile qτ 1nppq, for a given p P p1, 2s, as an
alternative risk measure to the quantile-VaR qαnp1q, as it allows to select the value τ 1n such that qτ 1nppq ” qαnp1q.
For instance, if the practitioner wishes to employ the expectile qτ 1np2q but still keep the probabilistic interpretation
of qαnp1q, Figure 9 (b) shows that the corresponding expectile level τ 1n :“ τ 1np2, αn; 1q may be approximated in the
present setup by its estimate pτ 1np2, αn; 1q “ 0.9999942.
If the interest now is in estimating the expectile q0.9999942p2q, one may wish to check how the Lp´quantile
estimators qqp0.9999942p2q in (13) and pqWpτ 1npp,0.9999942;2q
ppq in (16) differ from the benchmark estimators pqW0.9999942p2q
in (11) and rqW0.9999942p2q in (12) when the power p varies between 1 and 2. In Figure 10 (a) we plot the optimal
estimates p ÞÑ qqp0.9999942p2q in green, p ÞÑ pqWpτ 1npp,0.9999942;2q
ppq in blue, pqW0.9999942p2q in black, rqW0.9999942p2q in orange,
and the sample maximum in red. As is to be expected, the asymmetric least squares estimate pqW0.9999942p2q, in
black line, is clearly more pessimistic than the plug-in estimate rqW0.9999942p2q, in orange line, that heavily depends
on the optimistic Weissman quantile estimator pqW0.9999942p1q as can be seen from (12). The more sophisticated
expectile estimate p ÞÑ qqp0.9999942p2q, as green curve, steers overall a middle course behavior since it approaches
pqW0.9999942p2q as p tends to 2 and rqW0.9999942p2q when p tends to 1. By contrast, the composite expectile estimate
p ÞÑ pqWpτ 1npp,0.9999942;2q
ppq, as blue curve, appears to be the most conservative risk measure, especially as p decays to
1. The evolution of the extrapolated estimator pτ 1npp, 0.9999942; 2q in (15) of τ 1npp, 0.9999942; 2q in (14) is plotted in
Figure 10 (b) against p in blue, along with the expectile level 0.9999942 in red line.
References
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of extremes: Theory and applications,
Wiley.
Bingham, N. H., Goldie, C. M., Teugels, J. L. (1987). Regular Variation, Cambridge University Press, Cambridge.
Geyer, C.J. (1996). On the asymptotics of convex stochastic optimization, unpublished manuscript.
de Haan, L., Ferreira, A. (2006). Extreme Value Theory: An Introduction, Springer-Verlag, New York.
Ibragimov, I.A. (1962). Some limit theorems for stationary processes, Theory of Probability and its Applications
7(4): 349–382.
Lehmann, E.L. (1966). Some concepts of dependence, Annals of Mathematical Statistics 37(5): 1137–1153.
Utev, S.A. (1990). On the central limit theorem for ϕ´mixing arrays of random variables, Theory of Probability
and its Applications 35(1): 131–139.
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0
5000
10000
15000
10 12 14log(Claims)
count
0
4000
8000
12000
16000Count
(a)
4000000
4200000
4400000
1.00 1.25 1.50 1.75 2.00p
Paths
variablemaximum
composite
traditional
(b)
0.999990
0.999991
0.999992
0.999993
0.999994
1.00 1.25 1.50 1.75 2.00p
Plots
variabletau
alpha
(c)
Figure 9: Medical insurance data. (a)—Scatterplot and histogram. (b)—Composite estimates p ÞÑ pqWpτ 1npp,αn;1q
ppq in
blue, along with the traditional estimate pqWαnp1q in green and the sample maximum Xn,n in red. (c)—Extrapolated
estimates p ÞÑ pτ 1npp, αn; 1q in blue, along with the tail probability αn in red line.
38
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3800000
4000000
4200000
4400000
1.00 1.25 1.50 1.75 2.00p
Paths
variablemaximum
composite
check
hat
tilde
(a)
0.999990
0.999991
0.999992
0.999993
0.999994
1.00 1.25 1.50 1.75 2.00p
Plots
variabletau
alpha
(b)
Figure 10: (a)—The paths p ÞÑ qqp0.9999942p2q in green and p ÞÑ pqWpτ 1npp,0.9999942;2q
ppq in blue, along with pqW0.9999942p2q
in black, rqW0.9999942p2q in orange, and the sample maximum Xn,n in red. (b)—Extrapolated estimates p ÞÑ
pτ 1npp, 0.9999942; 2q in blue, along with the expectile level 0.9999942 in red line.
39
![Laurent.Gardes@inrialpes.fr arXiv:1104.0166v1 [math.ST] 1 ... · arXiv:1104.0166v1 [math.ST] 1 Apr 2011 FUNCTIONAL NONPARAMETRIC ESTIMATION OF CONDITIONAL EXTREME QUANTILES Laurent](https://static.documents.pub/doc/80x56/5b8b6e8909d3f227638b6772/inrialpesfr-arxiv11040166v1-mathst-1-arxiv11040166v1-mathst.jpg)
